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Theorem elrabsf 2947
 Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2838 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
elrabsf.1
Assertion
Ref Expression
elrabsf

Proof of Theorem elrabsf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2911 . 2
2 elrabsf.1 . . 3
3 nfcv 2281 . . 3
4 nfv 1508 . . 3
5 nfsbc1v 2927 . . 3
6 sbceq1a 2918 . . 3
72, 3, 4, 5, 6cbvrab 2684 . 2
81, 7elrab2 2843 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wcel 1480  wnfc 2268  crab 2420  wsbc 2909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-sbc 2910 This theorem is referenced by:  mpoxopovel  6138  zsupcllemstep  11645  infssuzex  11649
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