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Theorem elabf 2750
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 2225 . . 3  |-  F/_ x A
3 elabf.1 . . 3  |-  F/ x ps
4 elabf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
52, 3, 4elabgf 2749 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
61, 5ax-mp 7 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287   F/wnf 1392    e. wcel 1436   {cab 2071   _Vcvv 2615
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  elab  2751  indpi  6837
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