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Theorem elab3gf 3255
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3248. (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 3248 . . . 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 3248 . 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 1379   F/wnf 1599    e. wcel 1767   {cab 2452   F/_wnfc 2615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115
This theorem is referenced by:  elab3g  3256
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