Theorem cbvralvw 2790

Description: Version of cbvralv 2786 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 2299	. . . 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 1975	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
5		df-ral 2533	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-ral 2533	. 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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wal 1400   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514  df-clel 2234  df-ral 2533

This theorem is used by:  cbvral2vw  2797  cc1  7637  zsupssdc  10689  hashfibc  11304  wrdind  11515  wrd2ind  11516  reuccatpfxs1  11540  prmpwdvds  13162  nninfdclemcl  13396  grpinvalem  13763  grpinva  13764  issubg4m  14054  isnsg2  14064  elnmz  14069  fsumdvdsmul  16302  2sqlem6  16461  2sqlem10  16466  uspgr2wlkeq  16828  depindlem1  16969  bj-charfunbi  17059