Theorem cbvrexw 2736

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2732 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 1531 and ax-bndl 1533 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 2350	. 2 |- F/_ x A
2		nfcv 2350	. 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 2732	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   E.wrex 2487

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492

This theorem is referenced by:  cbvreuw  2737  reu8nf  3087  elabrexg  5850