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Theorem elab3gf 3248
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3241. (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 3241 . . . 4  |-  ( A  e.  { x  | 
ph }  ->  ( A  e.  { x  |  ph }  <->  ps )
)
54ibi 241 . . 3  |-  ( A  e.  { x  | 
ph }  ->  ps )
6 pm2.21 108 . . 3  |-  ( -. 
ps  ->  ( ps  ->  A  e.  { x  | 
ph } ) )
75, 6impbid2 204 . 2  |-  ( -. 
ps  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
81, 2, 3elabgf 3241 . 2  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
97, 8ja 161 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1398   F/wnf 1621    e. wcel 1823   {cab 2439   F/_wnfc 2602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108
This theorem is referenced by:  elab3g  3249
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