Theorem ceqsexv 2728

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 1509	. 2 |- F/ x ps
2		ceqsexv.1	. 2 |- A e. _V
3		ceqsexv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsex 2727	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689

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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691

This theorem is referenced by:  ceqsex3v  2731  gencbvex  2735  sbhypf  2738  euxfr2dc  2873  inuni  4088  eqvinop  4173  onm  4331  uniuni  4380  opeliunxp  4602  elvvv  4610  rexiunxp  4689  imai  4903  coi1  5062  abrexco  5668  opabex3d  6027  opabex3  6028  mapsnen  6713  xpsnen  6723  xpcomco  6728  xpassen  6732