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Theorem elrabsf 3029
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2922 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 2993 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 elrabsf.1 . . 3  |-  F/_ x B
3 nfcv 2419 . . 3  |-  F/_ y B
4 nfv 1605 . . 3  |-  F/ y
ph
5 nfsbc1v 3010 . . 3  |-  F/ x [. y  /  x ]. ph
6 sbceq1a 3001 . . 3  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
72, 3, 4, 5, 6cbvrab 2786 . 2  |-  { x  e.  B  |  ph }  =  { y  e.  B  |  [. y  /  x ]. ph }
81, 7elrab2 2925 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   F/_wnfc 2406   {crab 2547   [.wsbc 2991
This theorem is referenced by:  onminesb  4589  ac6num  8106  tfisg  24204  wfisg  24209  frinsg  24245  rabrenfdioph  26897  mpt2xopovel  28086  bnj23  28744  bnj1204  29042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992
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