Theorem ceqsexv 2861

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 1581	. 2 |- F/ x ps
2		ceqsexv.1	. 2 |- A e. _V
3		ceqsexv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsex 2860	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821

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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  ceqsex3v  2865  gencbvex  2869  sbhypf  2872  euxfr2dc  3011  inuni  4286  eqvinop  4378  onm  4541  uniuni  4592  opeliunxp  4825  elvvv  4833  rexiunxp  4917  imai  5138  coi1  5298  abrexco  5955  opabex3d  6340  opabex3  6341  mapsnen  7090  xpsnen  7109  xpcomco  7114  xpassen  7118