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Basic facts about Prop as a type

An intuitionistic theorem from topos theory [LambekScott]
References:
[LambekScott] Jim Lambek, Phil J. Scott, Introduction to higher order categorical logic, Cambridge Studies in Advanced Mathematics (Book 7), 1988. 
f:Prop -> Prop
inj:forall A B : Prop, f A <-> f B -> A <-> B
ext:forall A B : Prop, A <-> B -> f A <-> f B
forall A : Prop, f (f A) <-> A


f:Prop -> Prop
inj:forall A B : Prop, f A <-> f B -> A <-> B
ext:forall A B : Prop, A <-> B -> f A <-> f B
forall A : Prop, f (f A) <-> A


f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
f (f A) <-> A


f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
f (f (f A)) <-> f A


f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
f A
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
f (f (f A))


f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
f A
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
f A

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
f A <-> True

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
f (f A) <-> f True

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f (f A)
f True
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f True
f (f A)

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f (f A)
f True
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f (f A)
f True

    
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f (f A)
f (f (f A)) <-> f True

    apply ext; firstorder.
  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f True
f (f A)
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f True
f (f A)

    
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f (f (f A))
H':f True
f (f A) <-> True

    apply inj; firstorder.

f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
f (f (f A))
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
f (f (f A))

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
f A <-> f (f (f A))

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
A <-> f (f A)

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':A
f (f A)
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':f (f A)
A

  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':A
f (f A)
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':A
f (f A)

    
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':A
f A <-> f (f A)

    apply ext; firstorder.
  
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':f (f A)
A
 
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':f (f A)
A

    
f:Prop -> Prop
inj:forall A0 B : Prop, f A0 <-> f B -> A0 <-> B
ext:forall A0 B : Prop, A0 <-> B -> f A0 <-> f B
A:Prop
H:f A
H':f (f A)
f A <-> A

    apply inj; firstorder.
Defined.