Theorem cbv2h 2046

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-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 193	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 31	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	1, 2, 5	cbv1h 2045	. 2 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7		equcomi 1817	. . . . 5 |- ( y = x -> x = y )
8		biimpr 198	. . . . 5 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
9	7, 3, 8	syl56 32	. . . 4 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
10	2, 1, 9	cbv1h 2045	. . 3 |- ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11	10	alcoms 1867	. 2 |- ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12	6, 11	impbid 190	1 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638

This theorem is referenced by:  cbv2  2047  eujustALT  2241