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Theorem elab3gf 3324
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3316. (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 3316 . . . 4 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
54ibi 254 . . 3 (𝐴 ∈ {𝑥𝜑} → 𝜓)
6 pm2.21 118 . . 3 𝜓 → (𝜓𝐴 ∈ {𝑥𝜑}))
75, 6impbid2 214 . 2 𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
81, 2, 3elabgf 3316 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
97, 8ja 171 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wnf 1698  wcel 1976  {cab 2595  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174
This theorem is referenced by:  elab3g  3325
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