Theorem cbvrex 2698

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   y, A

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

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		nfcv 2317	. 2 |- F/_ x A
2		nfcv 2317	. 2 |- F/_ y A
3		cbvral.1	. 2 |- F/ y ph
4		cbvral.2	. 2 |- F/ x ps
5		cbvral.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrexf 2695	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1458   E.wrex 2454

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459

This theorem is referenced by:  cbvrmo  2700  cbvrexv  2702  cbvrexsv  2718  cbviun  3919  disjiun  3993  rexxpf  4767  isarep1  5294  rexrnmpt  5651  elabrex  5749