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Theorem elabf 2894
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabf.1  |-  F/ x ps
elabf.2  |-  A  e. 
_V
elabf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabf  |-  ( A  e.  { x  | 
ph }  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem elabf
StepHypRef Expression
1 elabf.2 . 2  |-  A  e. 
_V
2 nfcv 2331 . . 3  |-  F/_ x A
3 elabf.1 . . 3  |-  F/ x ps
4 elabf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
52, 3, 4elabgf 2893 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
61, 5ax-mp 5 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1363   F/wnf 1470    e. wcel 2159   {cab 2174   _Vcvv 2751
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753
This theorem is referenced by:  elab  2895  indpi  7358
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