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Theorem elrabsf 3826
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3680 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 3780 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2904 . . 3 𝑦𝐵
4 nfv 1918 . . 3 𝑦𝜑
5 nfsbc1v 3798 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 3789 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrabw 3468 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 3687 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  wnfc 2884  {crab 3433  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3779
This theorem is referenced by:  frpoinsg  6345  wfisgOLD  6356  onminesb  7781  tfisg  7843  mpoxopovel  8205  frinsg  9746  ac6num  10474  hashrabsn1  14334  bnj23  33729  bnj1204  34023  rabrenfdioph  41552
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