Theorem ceqsexgv 2725

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Hypothesis

Ref	Expression
ceqsexgv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem ceqsexgv

Step	Hyp	Ref	Expression
1		nfv 1462	. 2 |- F/ x ps
2		ceqsexgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	ceqsexg 2724	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434

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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604

This theorem is referenced by:  ceqsrexv  2726  clel3g  2730  elxp4  4832  elxp5  4833  dmfco  5267  fndmdif  5298  fndmin  5300  fmptco  5356  rexrnmpt2  5641  brtpos2  5894  xpsnen  6355  prarloc  6744