Theorem cbv2 2270

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (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		cbv2.1	. . 3 |- F/ x ph
2		cbv2.2	. . . 4 |- F/ y ph
3	2	nf5ri 2065	. . 3 |- ( ph -> A. y ph )
4	1, 3	alrimi 2082	. 2 |- ( ph -> A. x A. y ph )
5		cbv2.3	. . . 4 |- ( ph -> F/ y ps )
6	5	nf5rd 2066	. . 3 |- ( ph -> ( ps -> A. y ps ) )
7		cbv2.4	. . . 4 |- ( ph -> F/ x ch )
8	7	nf5rd 2066	. . 3 |- ( ph -> ( ch -> A. x ch ) )
9		cbv2.5	. . 3 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
10	6, 8, 9	cbv2h 2269	. 2 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )
11	4, 10	syl 17	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbvald  2277  sb9  2426  wl-cbvalnaed  33319  wl-sb8t  33333