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Theorem elrabf 2696
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 2566 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → 𝐴 ∈ V)
2 elex 2566 . . 3 (𝐴𝐵𝐴 ∈ V)
32adantr 261 . 2 ((𝐴𝐵𝜓) → 𝐴 ∈ V)
4 df-rab 2315 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
54eleq2i 2104 . . 3 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
6 elrabf.1 . . . 4 𝑥𝐴
7 elrabf.2 . . . . . 6 𝑥𝐵
86, 7nfel 2186 . . . . 5 𝑥 𝐴𝐵
9 elrabf.3 . . . . 5 𝑥𝜓
108, 9nfan 1457 . . . 4 𝑥(𝐴𝐵𝜓)
11 eleq1 2100 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
12 elrabf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
1311, 12anbi12d 442 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
146, 10, 13elabgf 2685 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
155, 14syl5bb 181 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓)))
161, 3, 15pm5.21nii 620 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wnf 1349  wcel 1393  {cab 2026  wnfc 2165  {crab 2310  Vcvv 2557
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
This theorem is referenced by:  elrab  2698  frind  4089  rabxfrd  4201
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