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Theorem elrabsf 3802
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3662 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 𝑥𝐵
Assertion
Ref Expression
elrabsf (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))

Proof of Theorem elrabsf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3760 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2982 . . 3 𝑦𝐵
4 nfv 1916 . . 3 𝑦𝜑
5 nfsbc1v 3778 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 3769 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrabw 3475 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 3669 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2115  wnfc 2962  {crab 3137  [wsbc 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-sbc 3759
This theorem is referenced by:  wfisg  6170  onminesb  7507  mpoxopovel  7882  ac6num  9899  hashrabsn1  13740  bnj23  32048  bnj1204  32344  tfisg  33115  frpoinsg  33141  frinsg  33147  rabrenfdioph  39675
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