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  7631  zsupssdc  10673  hashfibc  11283  wrdind  11494  wrd2ind  11495  reuccatpfxs1  11519  prmpwdvds  13134  nninfdclemcl  13339  grpinvalem  13705  grpinva  13706  issubg4m  13996  isnsg2  14006  elnmz  14011  fsumdvdsmul  16105  2sqlem6  16239  2sqlem10  16244  uspgr2wlkeq  16606  depindlem1  16747  bj-charfunbi  16837