Theorem cbv 180

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

Hypotheses

Ref	Expression
cbv.1	|- A:be
cbv.2	|- [x:al = y:al] |= [A = B]

Assertion

Ref	Expression
cbv	|- T. |= [\x:al A = \y:al B]

Distinct variable groups:   y,A   x,B   x,y,al   be,y

Proof of Theorem cbv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbv.1	. 2 |- A:be
2		wv 64	. . 3 |- z:al:al
3	1, 2	ax-17 105	. 2 |- T. |= [(\y:al Az:al) = A]
4		cbv.2	. . . 4 |- [x:al = y:al] |= [A = B]
5	1, 4	eqtypi 78	. . 3 |- B:be
6	5, 2	ax-17 105	. 2 |- T. |= [(\x:al Bz:al) = B]
7	1, 3, 6, 4	cbvf 179	1 |- T. |= [\x:al A = \y:al B]

Syntax hints:  tv 1  \kl 6   = ke 7  T.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