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Theorem ceqsexgv 2913
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 1609 . 2  |-  F/ x ps
2 ceqsexgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2ceqsexg 2912 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  ceqsrexv  2914  clel3g  2918  elxp5  5177  xpsnen  6962  isssc  13713  isgrpo  20879  ismgm  21003  islatalg  25286  ceqsex3vOLD  26830  pmapjat1  30664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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