Theorem cbvrexvw 2702

Description: Version of cbvrexv 2698 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 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 2232	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 471	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	cbvexvw 1914	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
5		df-rex 2455	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		df-rex 2455	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
7	4, 5, 6	3bitr4i 211	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1486   e. wcel 2142   E.wrex 2450

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 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-clel 2167  df-rex 2455

This theorem is referenced by:  cbvrex2vw  2709  prodmodclem2  11544  prodmodc  11545  zsupssdc  11913  pceu  12253  grpridd  12645  dfgrp2  12736  dfgrp3mlem  12801  bj-charfunbi  13931