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Theorem elrabf 3264
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 3127 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  A  e.  _V )
2 elex 3127 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
32adantr 465 . 2  |-  ( ( A  e.  B  /\  ps )  ->  A  e. 
_V )
4 df-rab 2826 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
54eleq2i 2545 . . 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 2642 . . . . 5  |-  F/ x  A  e.  B
9 elrabf.3 . . . . 5  |-  F/ x ps
108, 9nfan 1875 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
11 eleq1 2539 . . . . 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 3253 . . 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 1379   F/wnf 1599    e. wcel 1767   {cab 2452   F/_wnfc 2615   {crab 2821   _Vcvv 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120
This theorem is referenced by:  elrab  3266  rabasiun  4335  rabxfrd  4674  onminsb  6629  nnawordex  7298  tskwe  8343  rabssnn0fi  12075  iundisj  21826  iundisjf  27262  iundisjfi  27415  sltval2  29334  nobndlem5  29374  rfcnpre3  31301  rfcnpre4  31302  stoweidlem26  31640  stoweidlem27  31641  stoweidlem31  31645  stoweidlem34  31648  stoweidlem51  31665  stoweidlem52  31666  stoweidlem59  31673  fourierdlem20  31741  fourierdlem79  31800  bnj1388  33524
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