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Theorem elab3gf 2805
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2798. (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 2798 . . 3 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 175 . 2 (𝐴 ∈ {𝑥𝜑} → 𝜓)
61, 2, 3elabgf 2798 . . . 4 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
76imim2i 12 . . 3 ((𝜓𝐴𝐵) → (𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
8 bi2 129 . . 3 ((𝐴 ∈ {𝑥𝜑} ↔ 𝜓) → (𝜓𝐴 ∈ {𝑥𝜑}))
97, 8syli 37 . 2 ((𝜓𝐴𝐵) → (𝜓𝐴 ∈ {𝑥𝜑}))
105, 9impbid2 142 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnf 1419  wcel 1463  {cab 2101  wnfc 2243
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  elab3g  2806
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