Theorem cbvreuv 2694

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
cbvralv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   y, A   ph, y   ps, x

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

Proof of Theorem cbvreuv

Step	Hyp	Ref	Expression
1		nfv 1516	. 2 |- F/ y ph
2		nfv 1516	. 2 |- F/ x ps
3		cbvralv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvreu 2690	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   E!wreu 2446

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-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-cleq 2158  df-clel 2161  df-reu 2451

This theorem is referenced by:  reu8  2922