Theorem cbvrexw 2721

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2717 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 1518 and ax-bndl 1520 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 2336	. 2 |- F/_ x A
2		nfcv 2336	. 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 2717	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   E.wrex 2473

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478

This theorem is referenced by:  elabrexg  5801