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Theorem elab3g 3195
Description: Membership in a class abstraction, with a weaker antecedent than elabg 3190. (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 2610 . 2  |-  F/_ x A
2 nfv 1674 . 2  |-  F/ x ps
3 elab3g.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elab3gf 3194 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1757   {cab 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-v 3056
This theorem is referenced by:  elab3  3196  elssabg  4531  elrnmptg  5173  elrelimasn  5277  elmapg  7313  isust  19880  ellimc  21450  isismty  28824
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