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Theorem elab3g 2954
Description: Membership in a class abstraction, with a weaker antecedent than elabg 2949. (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 2452 . 2  |-  F/_ x A
2 nfv 1610 . 2  |-  F/ x ps
3 elab3g.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elab3gf 2953 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   {cab 2302
This theorem is referenced by:  elab3  2955  elssabg  4203  elrnmptg  4966  elrelimasn  5074  elmapg  6828  ellimc  19276  isust  23420  isismty  25673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824
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