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Theorem elab3g 3088
Description: Membership in a class abstraction, with a weaker antecedent than elabg 3083. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elab3g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab3g  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elab3g
StepHypRef Expression
1 nfcv 2572 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 elab3g.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elab3gf 3087 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422
This theorem is referenced by:  elab3  3089  elssabg  4355  elrnmptg  5120  elrelimasn  5228  elmapg  7031  isust  18233  ellimc  19760  isismty  26510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
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