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Theorem elrabf 3255
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 3118 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  A  e.  _V )
2 elex 3118 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
32adantr 465 . 2  |-  ( ( A  e.  B  /\  ps )  ->  A  e. 
_V )
4 df-rab 2816 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
54eleq2i 2535 . . 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 2632 . . . . 5  |-  F/ x  A  e.  B
9 elrabf.3 . . . . 5  |-  F/ x ps
108, 9nfan 1929 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
11 eleq1 2529 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
12 elrabf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
146, 10, 13elabgf 3244 . . 3  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ( x  e.  B  /\  ph ) } 
<->  ( A  e.  B  /\  ps ) ) )
155, 14syl5bb 257 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) ) )
161, 3, 15pm5.21nii 353 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   F/wnf 1617    e. wcel 1819   {cab 2442   F/_wnfc 2605   {crab 2811   _Vcvv 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111
This theorem is referenced by:  elrab  3257  rabasiun  4336  invdisjrab  4446  rabxfrd  4677  onminsb  6633  nnawordex  7304  tskwe  8348  rabssnn0fi  12098  iundisj  22084  rabtru  27527  iundisjf  27588  iundisjfi  27761  sltval2  29633  nobndlem5  29673  rfcnpre3  31611  rfcnpre4  31612  stoweidlem26  32011  stoweidlem27  32012  stoweidlem31  32016  stoweidlem34  32019  stoweidlem51  32036  stoweidlem52  32037  stoweidlem59  32044  fourierdlem20  32112  fourierdlem79  32171  bnj1388  34232
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