Theorem cbvralvw 2730

Description: Version of cbvralv 2726 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
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 2254	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	imbi12d 234	. . 3 |- ( x = y -> ( ( x e. A -> ph ) <-> ( y e. A -> ps ) ) )
4	3	cbvalvw 1931	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
5		df-ral 2477	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-ral 2477	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
7	4, 5, 6	3bitr4i 212	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   e. wcel 2164   A.wral 2472

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-nf 1472  df-clel 2189  df-ral 2477

This theorem is referenced by:  cbvral2vw  2737  cc1  7325  zsupssdc  12091  prmpwdvds  12493  nninfdclemcl  12605  grpinvalem  12968  grpinva  12969  issubg4m  13263  isnsg2  13273  elnmz  13278  2sqlem6  15207  2sqlem10  15212  bj-charfunbi  15303