Theorem cbvrexvw 2731

Description: Version of cbvrexv 2727 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvralvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrexvw	|- ( E. x e. A ph <-> E. y e. A ps )

Distinct variable groups:   x, y, A   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvrexvw

Step	Hyp	Ref	Expression
1		eleq1w 2254	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 473	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	cbvexvw 1932	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
5		df-rex 2478	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		df-rex 2478	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
7	4, 5, 6	3bitr4i 212	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1503   e. wcel 2164   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-clel 2189  df-rex 2478

This theorem is referenced by:  cbvrex2vw  2738  prodmodclem2  11720  prodmodc  11721  zsupssdc  12091  pceu  12433  4sqlem12  12540  nninfdclemcl  12605  grprida  12970  dfgrp2  13099  dfgrp3mlem  13170  lss1d  13879  bj-charfunbi  15303