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Theorem elabgf 2800
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 2260 . . . 4  |-  F/_ x { x  |  ph }
31, 2nfel 2267 . . 3  |-  F/ x  A  e.  { x  |  ph }
4 elabgf.2 . . 3  |-  F/ x ps
53, 4nfbi 1553 . 2  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
6 eleq1 2180 . . 3  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
7 elabgf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7bibi12d 234 . 2  |-  ( x  =  A  ->  (
( x  e.  {
x  |  ph }  <->  ph )  <->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
9 abid 2105 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
101, 5, 8, 9vtoclgf 2718 1  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   F/wnf 1421    e. wcel 1465   {cab 2103   F/_wnfc 2245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  elabf  2801  elabg  2803  elab3gf  2807  elrabf  2811  bj-intabssel  12923
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