Theorem cbv2h 1705

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbv2h.1	|- ( ph -> ( ps -> A. y ps ) )
cbv2h.2	|- ( ph -> ( ch -> A. x ch ) )
cbv2h.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
cbv2h	|- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Proof of Theorem cbv2h

Step	Hyp	Ref	Expression
1		cbv2h.1	. . 3 |- ( ph -> ( ps -> A. y ps ) )
2		cbv2h.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3		cbv2h.3	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		bi1 117	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 33	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	1, 2, 5	cbv1h 1704	. 2 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7		equcomi 1661	. . . . 5 |- ( y = x -> x = y )
8		bi2 129	. . . . 5 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
9	7, 3, 8	syl56 34	. . . 4 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
10	2, 1, 9	cbv1h 1704	. . 3 |- ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11	10	a7s 1411	. 2 |- ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12	6, 11	impbid 128	1 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496

This theorem depends on definitions:  df-bi 116  df-nf 1418

This theorem is referenced by:  cbv2  1706