Theorem cbv2h 1634

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 111	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 29	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	1, 2, 5	cbv1h 1633	. 2 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7		equcomi 1592	. . . . 5 |- ( y = x -> x = y )
8		bi2 121	. . . . 5 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
9	7, 3, 8	syl56 30	. . . 4 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
10	2, 1, 9	cbv1h 1633	. . 3 |- ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11	10	a7s 1343	. 2 |- ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12	6, 11	impbid 120	1 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbv2  1635