Theorem cbvrexw 2732

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2728 with more disjoint variable conditions. Although we don't do so yet, we expect the disjoint variable conditions will allow us to remove reliance on ax-i12 1529 and ax-bndl 1531 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
cbvrexw	|- ( E. x e. A ph <-> E. y e. A ps )

Distinct variable group:   x, y, A

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

Proof of Theorem cbvrexw

Step	Hyp	Ref	Expression
1		nfcv 2347	. 2 |- F/_ x A
2		nfcv 2347	. 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	cbvrexfw 2728	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1482   E.wrex 2484

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489

This theorem is referenced by:  elabrexg  5826