Theorem cbv 180

Description: Change bound variables in a lambda abstraction. (Contributed by Mario Carneiro, 8-Oct-2014.)

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 64	. . 3 ⊢ z:α:α
3	1, 2	ax-17 105	. 2 ⊢ ⊤⊧[(λy:α Az:α) = A]
4		cbv.2	. . . 4 ⊢ [x:α = y:α]⊧[A = B]
5	1, 4	eqtypi 78	. . 3 ⊢ B:β
6	5, 2	ax-17 105	. 2 ⊢ ⊤⊧[(λx:α Bz:α) = B]
7	1, 3, 6, 4	cbvf 179	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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126

This theorem is referenced by:  ax10  213