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Theorem elabgf 2913
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 2422 . . . 4  |-  F/_ x { x  |  ph }
31, 2nfel 2428 . . 3  |-  F/ x  A  e.  { x  |  ph }
4 elabgf.2 . . 3  |-  F/ x ps
53, 4nfbi 1773 . 2  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
6 eleq1 2344 . . 3  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
7 elabgf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7bibi12d 314 . 2  |-  ( x  =  A  ->  (
( x  e.  {
x  |  ph }  <->  ph )  <->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
9 abid 2272 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
101, 5, 8, 9vtoclgf 2843 1  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1532    = wceq 1624    e. wcel 1685   {cab 2270   F/_wnfc 2407
This theorem is referenced by:  elabf  2914  elabg  2916  elab3gf  2920  elrabf  2923
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791
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