Theorem cbvrexvw 2785

Description: Version of cbvrexv 2781 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 2295	. . . 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 1972	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
5		df-rex 2528	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		df-rex 2528	. 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 1541   e. wcel 2205   E.wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-clel 2230  df-rex 2528

This theorem is referenced by:  cbvrex2vw  2792  zsupssdc  10622  prodmodclem2  12288  prodmodc  12289  pceu  13018  4sqlem12  13125  nninfdclemcl  13283  grprida  13650  dfgrp2  13782  dfgrp3mlem  13853  lss1d  14657  2lgslem1b  16088  bj-charfunbi  16707