Theorem cbvreuv 2720

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 1539	. 2 |- F/ y ph
2		nfv 1539	. 2 |- F/ x ps
3		cbvralv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvreu 2716	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   E!wreu 2470

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-cleq 2182  df-clel 2185  df-reu 2475

This theorem is referenced by:  reu8  2948