Theorem cbv1h 1734

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)

Hypotheses

Ref	Expression
cbv1h.1	|- ( ph -> ( ps -> A. y ps ) )
cbv1h.2	|- ( ph -> ( ch -> A. x ch ) )
cbv1h.3	|- ( ph -> ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
cbv1h	|- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )

Proof of Theorem cbv1h

Step	Hyp	Ref	Expression
1		nfa1 1529	. 2 |- F/ x A. x A. y ph
2		nfa2 1567	. 2 |- F/ y A. x A. y ph
3		sp 1499	. . . . 5 |- ( A. y ph -> ph )
4	3	sps 1525	. . . 4 |- ( A. x A. y ph -> ph )
5		cbv1h.1	. . . 4 |- ( ph -> ( ps -> A. y ps ) )
6	4, 5	syl 14	. . 3 |- ( A. x A. y ph -> ( ps -> A. y ps ) )
7	2, 6	nfd 1511	. 2 |- ( A. x A. y ph -> F/ y ps )
8		cbv1h.2	. . . 4 |- ( ph -> ( ch -> A. x ch ) )
9	4, 8	syl 14	. . 3 |- ( A. x A. y ph -> ( ch -> A. x ch ) )
10	1, 9	nfd 1511	. 2 |- ( A. x A. y ph -> F/ x ch )
11		cbv1h.3	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
12	4, 11	syl 14	. 2 |- ( A. x A. y ph -> ( x = y -> ( ps -> ch ) ) )
13	1, 2, 7, 10, 12	cbv1 1733	1 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )

Syntax hints:   -> wi 4   A.wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  cbv2h  1736