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Theorem elab3gf 2751
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2744. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1 𝑥𝐴
elab3gf.2 𝑥𝜓
elab3gf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab3gf ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Proof of Theorem elab3gf
StepHypRef Expression
1 elab3gf.1 . . . 4 𝑥𝐴
2 elab3gf.2 . . . 4 𝑥𝜓
3 elab3gf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabgf 2744 . . 3 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 174 . 2 (𝐴 ∈ {𝑥𝜑} → 𝜓)
61, 2, 3elabgf 2744 . . . 4 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
76imim2i 12 . . 3 ((𝜓𝐴𝐵) → (𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
8 bi2 128 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜓) → (𝜓𝐴 ∈ {𝑥𝜑}))
97, 8syli 37 . 2 ((𝜓𝐴𝐵) → (𝜓𝐴 ∈ {𝑥𝜑}))
105, 9impbid2 141 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wnf 1390  wcel 1434  {cab 2069  wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by:  elab3g  2752
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