Theorem cbvralvw 2700

Description: Version of cbvralv 2696 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
cbvralvw	|- ( A. x e. A ph <-> A. y e. A ps )

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

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

Proof of Theorem cbvralvw

Step	Hyp	Ref	Expression
1		eleq1w 2231	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	imbi12d 233	. . 3 |- ( x = y -> ( ( x e. A -> ph ) <-> ( y e. A -> ps ) ) )
4	3	cbvalvw 1912	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
5		df-ral 2453	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-ral 2453	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
7	4, 5, 6	3bitr4i 211	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   e. wcel 2141   A.wral 2448

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-clel 2166  df-ral 2453

This theorem is referenced by:  cbvral2vw  2707  cc1  7227  zsupssdc  11909  prmpwdvds  12307  grprinvlem  12639  grprinvd  12640  2sqlem6  13750  2sqlem10  13755  bj-charfunbi  13846