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Theorem elrabsf 3813
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3673 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 3771 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2974 . . 3 𝑦𝐵
4 nfv 1906 . . 3 𝑦𝜑
5 nfsbc1v 3789 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 3780 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrabw 3487 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 3680 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  wnfc 2958  {crab 3139  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-sbc 3770
This theorem is referenced by:  wfisg  6176  onminesb  7502  mpoxopovel  7875  ac6num  9889  hashrabsn1  13723  bnj23  31887  bnj1204  32181  tfisg  32952  frpoinsg  32978  frinsg  32984  rabrenfdioph  39289
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