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Theorem elab3gf 3079
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3072. (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 3072 . . . 4  |-  ( A  e.  { x  | 
ph }  ->  ( A  e.  { x  |  ph }  <->  ps )
)
54ibi 233 . . 3  |-  ( A  e.  { x  | 
ph }  ->  ps )
6 pm2.21 102 . . 3  |-  ( -. 
ps  ->  ( ps  ->  A  e.  { x  | 
ph } ) )
75, 6impbid2 196 . 2  |-  ( -. 
ps  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
81, 2, 3elabgf 3072 . 2  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
97, 8ja 155 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   F/wnf 1553    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558
This theorem is referenced by:  elab3g  3080
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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