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Theorem elrabsf 3201
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3093 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 3165 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 elrabsf.1 . . 3  |-  F/_ x B
3 nfcv 2574 . . 3  |-  F/_ y B
4 nfv 1630 . . 3  |-  F/ y
ph
5 nfsbc1v 3182 . . 3  |-  F/ x [. y  /  x ]. ph
6 sbceq1a 3173 . . 3  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
72, 3, 4, 5, 6cbvrab 2956 . 2  |-  { x  e.  B  |  ph }  =  { y  e.  B  |  [. y  /  x ]. ph }
81, 7elrab2 3096 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1726   F/_wnfc 2561   {crab 2711   [.wsbc 3163
This theorem is referenced by:  onminesb  4780  mpt2xopovel  6473  ac6num  8361  tfisg  25481  wfisg  25486  frinsg  25522  rabrenfdioph  26877  bnj23  29085  bnj1204  29383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-sbc 3164
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