Theorem cbv2 1771

Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Hypotheses

Ref	Expression
cbv2.1	|- F/ x ph
cbv2.2	|- F/ y ph
cbv2.3	|- ( ph -> F/ y ps )
cbv2.4	|- ( ph -> F/ x ch )
cbv2.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
cbv2	|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.2	. . . 4 |- F/ y ph
2	1	nfri 1541	. . 3 |- ( ph -> A. y ph )
3		cbv2.1	. . . . 5 |- F/ x ph
4	3	nfal 1598	. . . 4 |- F/ x A. y ph
5	4	nfri 1541	. . 3 |- ( A. y ph -> A. x A. y ph )
6	2, 5	syl 14	. 2 |- ( ph -> A. x A. y ph )
7		cbv2.3	. . . 4 |- ( ph -> F/ y ps )
8	7	nfrd 1542	. . 3 |- ( ph -> ( ps -> A. y ps ) )
9		cbv2.4	. . . 4 |- ( ph -> F/ x ch )
10	9	nfrd 1542	. . 3 |- ( ph -> ( ch -> A. x ch ) )
11		cbv2.5	. . 3 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
12	8, 10, 11	cbv2h 1770	. 2 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )
13	6, 12	syl 14	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370   F/wnf 1482

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  cbvald  1948  cbvrald  15657