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Theorem elrabf 2718
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 2583 . 2 (𝐴 ∈ {𝑥𝐵𝜑} → 𝐴 ∈ V)
2 elex 2583 . . 3 (𝐴𝐵𝐴 ∈ V)
32adantr 265 . 2 ((𝐴𝐵𝜓) → 𝐴 ∈ V)
4 df-rab 2332 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
54eleq2i 2120 . . 3 (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)})
6 elrabf.1 . . . 4 𝑥𝐴
7 elrabf.2 . . . . . 6 𝑥𝐵
86, 7nfel 2202 . . . . 5 𝑥 𝐴𝐵
9 elrabf.3 . . . . 5 𝑥𝜓
108, 9nfan 1473 . . . 4 𝑥(𝐴𝐵𝜓)
11 eleq1 2116 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
12 elrabf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
1311, 12anbi12d 450 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
146, 10, 13elabgf 2707 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
155, 14syl5bb 185 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓)))
161, 3, 15pm5.21nii 630 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wcel 1409  {cab 2042  wnfc 2181  {crab 2327  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576
This theorem is referenced by:  elrab  2720  frind  4116  rabxfrd  4228
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