Theorem cbvralw 2723

Description: Rule used to change bound variables, using implicit substitution. Version of cbvral 2725 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1521 and ax-bndl 1523 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvralw.1	|- F/ y ph
cbvralw.2	|- F/ x ps
cbvralw.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvralw	|- ( A. x e. A ph <-> A. y e. A ps )

Distinct variable group:   x, y, A

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

Proof of Theorem cbvralw

Step	Hyp	Ref	Expression
1		nfcv 2339	. 2 |- F/_ x A
2		nfcv 2339	. 2 |- F/_ y A
3		cbvralw.1	. 2 |- F/ y ph
4		cbvralw.2	. 2 |- F/ x ps
5		cbvralw.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvralfw 2719	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1474   A.wral 2475

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  pcmptdvds  12514  lgseisenlem2  15312