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Theorem elabgf 2912
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1  |-  F/_ x A
elabgf.2  |-  F/ x ps
elabgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabgf  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2  |-  F/_ x A
2 nfab1 2421 . . . 4  |-  F/_ x { x  |  ph }
31, 2nfel 2427 . . 3  |-  F/ x  A  e.  { x  |  ph }
4 elabgf.2 . . 3  |-  F/ x ps
53, 4nfbi 1772 . 2  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
6 eleq1 2343 . . 3  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
7 elabgf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7bibi12d 312 . 2  |-  ( x  =  A  ->  (
( x  e.  {
x  |  ph }  <->  ph )  <->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
9 abid 2271 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
101, 5, 8, 9vtoclgf 2842 1  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406
This theorem is referenced by:  elabf  2913  elabg  2915  elab3gf  2919  elrabf  2922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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