Theorem cbvralvw 2772

Description: Version of cbvralv 2768 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 2292	. . . 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 1968	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
5		df-ral 2516	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-ral 2516	. 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 1396   e. wcel 2202   A.wral 2511

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-nf 1510  df-clel 2227  df-ral 2516

This theorem is referenced by:  cbvral2vw  2779  cc1  7544  zsupssdc  10561  wrdind  11369  wrd2ind  11370  reuccatpfxs1  11394  prmpwdvds  13008  nninfdclemcl  13149  grpinvalem  13548  grpinva  13549  issubg4m  13860  isnsg2  13870  elnmz  13875  fsumdvdsmul  15805  2sqlem6  15939  2sqlem10  15944  uspgr2wlkeq  16306  depindlem1  16447  bj-charfunbi  16527