Theorem cbv 168

Description: Change bound variables in a lambda abstraction.

Hypotheses

Ref	Expression
cbv.1	⊢ A:β
cbv.2	⊢ [x:α = y:α]⊧[A = B]

Assertion

Ref	Expression
cbv	⊢ ⊤⊧[λx:α A = λy:α B]

Distinct variable groups:   y,A   x,B   x,y,α   β,y

Proof of Theorem cbv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbv.1	. 2 ⊢ A:β
2		wv 58	. . 3 ⊢ z:α:α
3	1, 2	ax-17 95	. 2 ⊢ ⊤⊧[(λy:α Az:α) = A]
4		cbv.2	. . . 4 ⊢ [x:α = y:α]⊧[A = B]
5	1, 4	eqtypi 69	. . 3 ⊢ B:β
6	5, 2	ax-17 95	. 2 ⊢ ⊤⊧[(λx:α Bz:α) = B]
7	1, 3, 6, 4	cbvf 167	1 ⊢ ⊤⊧[λx:α A = λy:α B]

Syntax hints:  tv 1  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116

This theorem is referenced by:  ax10  200