Theorem cbvralw 2687

Description: Rule used to change bound variables, using implicit substitution. Version of cbvral 2688 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 1495 and ax-bndl 1497 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 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 2308	. 2 |- F/_ x A
2		nfcv 2308	. 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 2683	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  pcmptdvds  12275