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Theorem elrabsf 3824
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3678 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 3778 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2901 . . 3 𝑦𝐵
4 nfv 1915 . . 3 𝑦𝜑
5 nfsbc1v 3796 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 3787 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrabw 3465 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 3685 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2104  wnfc 2881  {crab 3430  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-sbc 3777
This theorem is referenced by:  frpoinsg  6343  wfisgOLD  6354  onminesb  7783  tfisg  7845  mpoxopovel  8207  frinsg  9748  ac6num  10476  hashrabsn1  14338  bnj23  34027  bnj1204  34321  rabrenfdioph  41854
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