ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsexgv Unicode version

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
StepHypRef Expression
1 nfv 1462 . 2  |-  F/ x ps
2 ceqsexgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2ceqsexg 2724 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator