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Theorem elrabf 2841
Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
Hypotheses
Ref Expression
elrabf.1 𝑥𝐴
elrabf.2 𝑥𝐵
elrabf.3 𝑥𝜓
elrabf.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elrabf (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))

Proof of Theorem elrabf
StepHypRef Expression
1 elex 2700 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → 𝐴 ∈ V)
2 elex 2700 . . 3 (𝐴𝐵𝐴 ∈ V)
32adantr 274 . 2 ((𝐴𝐵𝜓) → 𝐴 ∈ V)
4 df-rab 2426 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
54eleq2i 2207 . . 3 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
6 elrabf.1 . . . 4 𝑥𝐴
7 elrabf.2 . . . . . 6 𝑥𝐵
86, 7nfel 2291 . . . . 5 𝑥 𝐴𝐵
9 elrabf.3 . . . . 5 𝑥𝜓
108, 9nfan 1545 . . . 4 𝑥(𝐴𝐵𝜓)
11 eleq1 2203 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
12 elrabf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
1311, 12anbi12d 465 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
146, 10, 13elabgf 2829 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
155, 14syl5bb 191 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓)))
161, 3, 15pm5.21nii 694 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wnf 1437  wcel 1481  {cab 2126  wnfc 2269  {crab 2421  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691
This theorem is referenced by:  elrab  2843  frind  4281  rabxfrd  4397  infssuzcldc  11678
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