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Theorem elrabsf 2801
 Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 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 2766 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2178 . . 3 𝑦𝐵
4 nfv 1421 . . 3 𝑦𝜑
5 nfsbc1v 2782 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 2773 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrab 2555 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 2700 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  Ⅎwnfc 2165  {crab 2310  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-sbc 2765 This theorem is referenced by:  mpt2xopovel  5856
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