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Theorem elab3gf 2932
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2925. (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 2925 . . . 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 2925 . 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 1534    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419
This theorem is referenced by:  elab3g  2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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