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Theorem elrabf 2809
Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
Hypotheses
Ref Expression
elrabf.1  |-  F/_ x A
elrabf.2  |-  F/_ x B
elrabf.3  |-  F/ x ps
elrabf.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elrabf  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )

Proof of Theorem elrabf
StepHypRef Expression
1 elex 2669 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  A  e.  _V )
2 elex 2669 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
32adantr 272 . 2  |-  ( ( A  e.  B  /\  ps )  ->  A  e. 
_V )
4 df-rab 2400 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
54eleq2i 2182 . . 3  |-  ( A  e.  { x  e.  B  |  ph }  <->  A  e.  { x  |  ( x  e.  B  /\  ph ) } )
6 elrabf.1 . . . 4  |-  F/_ x A
7 elrabf.2 . . . . . 6  |-  F/_ x B
86, 7nfel 2265 . . . . 5  |-  F/ x  A  e.  B
9 elrabf.3 . . . . 5  |-  F/ x ps
108, 9nfan 1527 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
11 eleq1 2178 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
12 elrabf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 462 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
146, 10, 13elabgf 2798 . . 3  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ( x  e.  B  /\  ph ) } 
<->  ( A  e.  B  /\  ps ) ) )
155, 14syl5bb 191 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) ) )
161, 3, 15pm5.21nii 676 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   F/wnf 1419    e. wcel 1463   {cab 2101   F/_wnfc 2243   {crab 2395   _Vcvv 2658
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660
This theorem is referenced by:  elrab  2811  frind  4242  rabxfrd  4358  infssuzcldc  11540
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