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Theorem elab3gf 3669
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3661. (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 . . . . 5 𝑥𝐴
2 elab3gf.2 . . . . 5 𝑥𝜓
3 elab3gf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabgf 3661 . . . 4 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 268 . . 3 (𝐴 ∈ {𝑥𝜑} → 𝜓)
6 pm2.21 123 . . 3 𝜓 → (𝜓𝐴 ∈ {𝑥𝜑}))
75, 6impbid2 227 . 2 𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
81, 2, 3elabgf 3661 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
97, 8ja 187 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207   = wceq 1528  wnf 1775  wcel 2105  {cab 2796  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494
This theorem is referenced by:  elab3g  3670
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