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Theorem elab3g 3194
Description: Membership in a class abstraction, with a weaker antecedent than elabg 3189. (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 2558 . 2  |-  F/_ x A
2 nfv 1722 . 2  |-  F/ x ps
3 elab3g.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elab3gf 3193 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1836   {cab 2381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-v 3053
This theorem is referenced by:  elab3  3195  elssabg  4537  elrnmptg  5182  elrelimasn  5290  elmapg  7373  isust  20814  ellimc  22385  isismty  30503  frege55lem1c  38454
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