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Theorem elrabsf 3213
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3104 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 3177 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 elrabsf.1 . . 3  |-  F/_ x B
3 nfcv 2569 . . 3  |-  F/_ y B
4 nfv 1672 . . 3  |-  F/ y
ph
5 nfsbc1v 3194 . . 3  |-  F/ x [. y  /  x ]. ph
6 sbceq1a 3185 . . 3  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
72, 3, 4, 5, 6cbvrab 2960 . 2  |-  { x  e.  B  |  ph }  =  { y  e.  B  |  [. y  /  x ]. ph }
81, 7elrab2 3108 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1755   F/_wnfc 2556   {crab 2709   [.wsbc 3175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rab 2714  df-v 2964  df-sbc 3176
This theorem is referenced by:  onminesb  6398  mpt2xopovel  6726  ac6num  8636  tfisg  27512  wfisg  27517  frinsg  27553  rabrenfdioph  28998  hashrabsn1  30079  bnj23  31430  bnj1204  31726
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