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Theorem elab3gf 3505
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3497. (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 3497 . . . 4 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 256 . . 3 (𝐴 ∈ {𝑥𝜑} → 𝜓)
6 pm2.21 121 . . 3 𝜓 → (𝜓𝐴 ∈ {𝑥𝜑}))
75, 6impbid2 216 . 2 𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
81, 2, 3elabgf 3497 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
97, 8ja 174 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1630  wnf 1855  wcel 2144  {cab 2756  wnfc 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351
This theorem is referenced by:  elab3g  3506
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