Theorem cbv2h 1748

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		biimp 118	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 33	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	1, 2, 5	cbv1h 1746	. 2 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7		equcomi 1704	. . . . 5 |- ( y = x -> x = y )
8		biimpr 130	. . . . 5 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
9	7, 3, 8	syl56 34	. . . 4 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
10	2, 1, 9	cbv1h 1746	. . 3 |- ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11	10	a7s 1454	. 2 |- ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12	6, 11	impbid 129	1 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  cbv2  1749