MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrabf Structured version   Unicode version

Theorem elrabf 3084
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 2957 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  A  e.  _V )
2 elex 2957 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
32adantr 452 . 2  |-  ( ( A  e.  B  /\  ps )  ->  A  e. 
_V )
4 df-rab 2707 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
54eleq2i 2500 . . 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 2580 . . . . 5  |-  F/ x  A  e.  B
9 elrabf.3 . . . . 5  |-  F/ x ps
108, 9nfan 1846 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
11 eleq1 2496 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
12 elrabf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1311, 12anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
146, 10, 13elabgf 3073 . . 3  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ( x  e.  B  /\  ph ) } 
<->  ( A  e.  B  /\  ps ) ) )
155, 14syl5bb 249 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) ) )
161, 3, 15pm5.21nii 343 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   {cab 2422   F/_wnfc 2559   {crab 2702   _Vcvv 2949
This theorem is referenced by:  elrab  3085  rabxfrd  4737  onminsb  4772  nnawordex  6873  tskwe  7830  iundisj  19435  iundisjf  24022  iundisjfi  24145  sltval2  25604  nobndlem5  25644  rfcnpre3  27672  rfcnpre4  27673  stoweidlem26  27743  stoweidlem27  27744  stoweidlem31  27748  stoweidlem34  27751  stoweidlem51  27768  stoweidlem52  27769  stoweidlem59  27776  bnj1388  29340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951
  Copyright terms: Public domain W3C validator