Theorem ceqsexv 2638

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Hypotheses

Ref	Expression
ceqsexv.1	|- A e. _V
ceqsexv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsexv	|- ( E. x ( x = A /\ ph ) <-> ps )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsexv

Step	Hyp	Ref	Expression
1		nfv 1461	. 2 |- F/ x ps
2		ceqsexv.1	. 2 |- A e. _V
3		ceqsexv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsex 2637	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsex3v  2641  gencbvex  2645  sbhypf  2648  euxfr2dc  2777  inuni  3930  eqvinop  3998  onm  4156  uniuni  4201  opeliunxp  4413  elvvv  4421  rexiunxp  4496  imai  4701  coi1  4856  abrexco  5419  opabex3d  5768  opabex3  5769  xpsnen  6318  xpcomco  6323  xpassen  6327