Theorem ceqsexgv 2814

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 1508	. 2 |- F/ x ps
2		ceqsexgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	ceqsexg 2813	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  ceqsrexv  2815  clel3g  2819  elxp4  5026  elxp5  5027  dmfco  5489  fndmdif  5525  fndmin  5527  fmptco  5586  rexrnmpo  5886  brtpos2  6148  xpsnen  6715  prarloc  7318  metrest  12685