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Theorem elab3gf 2692
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2685. (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 . . . 4  |-  F/_ x A
2 elab3gf.2 . . . 4  |-  F/ x ps
3 elab3gf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2685 . . 3  |-  ( A  e.  { x  | 
ph }  ->  ( A  e.  { x  |  ph }  <->  ps )
)
54ibi 165 . 2  |-  ( A  e.  { x  | 
ph }  ->  ps )
61, 2, 3elabgf 2685 . . . 4  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
76imim2i 12 . . 3  |-  ( ( ps  ->  A  e.  B )  ->  ( ps  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 bi2 121 . . 3  |-  ( ( A  e.  { x  |  ph }  <->  ps )  ->  ( ps  ->  A  e.  { x  |  ph } ) )
97, 8syli 33 . 2  |-  ( ( ps  ->  A  e.  B )  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
105, 9impbid2 131 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243   F/wnf 1349    e. wcel 1393   {cab 2026   F/_wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  elab3g  2693
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