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Theorem elabg 2883
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
Hypothesis
Ref Expression
elabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabg  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elabg
StepHypRef Expression
1 nfcv 2319 . 2  |-  F/_ x A
2 nfv 1528 . 2  |-  F/ x ps
3 elabg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2879 1  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  elab2g  2884  intmin3  3871  finds  4599  elxpi  4642  ovelrn  6022  elfi  6969  indpi  7340  peano5nnnn  7890  peano5nni  8921  eltg  13488  eltg2  13489
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