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Theorem elrabsf 2877
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2769 has implicit substitution). The hypothesis specifies that 
x must not be a free variable in  B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
elrabsf.1  |-  F/_ x B
Assertion
Ref Expression
elrabsf  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )

Proof of Theorem elrabsf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2842 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 elrabsf.1 . . 3  |-  F/_ x B
3 nfcv 2228 . . 3  |-  F/_ y B
4 nfv 1466 . . 3  |-  F/ y
ph
5 nfsbc1v 2858 . . 3  |-  F/ x [. y  /  x ]. ph
6 sbceq1a 2849 . . 3  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
72, 3, 4, 5, 6cbvrab 2617 . 2  |-  { x  e.  B  |  ph }  =  { y  e.  B  |  [. y  /  x ]. ph }
81, 7elrab2 2774 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   F/_wnfc 2215   {crab 2363   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-sbc 2841
This theorem is referenced by:  mpt2xopovel  6006  zsupcllemstep  11215  infssuzex  11219
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