MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsexgv Structured version   Unicode version

Theorem ceqsexgv 3216
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 1692 . 2  |-  F/ x ps
2 ceqsexgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2ceqsexg 3215 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-v 3095
This theorem is referenced by:  ceqsrexv  3217  clel3g  3221  elxp5  6726  xpsnen  7599  isssc  15061  metuel2  20948  isgrpo  25063  ismgmOLD  25187  ceqsex3vOLD  30570  bj-finsumval0  34365  pmapjat1  35279
  Copyright terms: Public domain W3C validator