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Theorem elab3gf 2919
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2912. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1  |-  F/_ x A
elab3gf.2  |-  F/ x ps
elab3gf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab3gf  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)

Proof of Theorem elab3gf
StepHypRef Expression
1 elab3gf.1 . . . . 5  |-  F/_ x A
2 elab3gf.2 . . . . 5  |-  F/ x ps
3 elab3gf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2912 . . . 4  |-  ( A  e.  { x  | 
ph }  ->  ( A  e.  { x  |  ph }  <->  ps )
)
54ibi 232 . . 3  |-  ( A  e.  { x  | 
ph }  ->  ps )
6 pm2.21 100 . . 3  |-  ( -. 
ps  ->  ( ps  ->  A  e.  { x  | 
ph } ) )
75, 6impbid2 195 . 2  |-  ( -. 
ps  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
81, 2, 3elabgf 2912 . 2  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
97, 8ja 153 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406
This theorem is referenced by:  elab3g  2920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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