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Theorem elab3g 3684
Description: Membership in a class abstraction, with a weaker antecedent than elabg 3675. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elab3g.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab3g ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab3g
StepHypRef Expression
1 elab3g.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21elabg 3675 . . . 4 (𝐴 ∈ {𝑥𝜑} → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
32ibi 267 . . 3 (𝐴 ∈ {𝑥𝜑} → 𝜓)
4 pm2.21 123 . . 3 𝜓 → (𝜓𝐴 ∈ {𝑥𝜑}))
53, 4impbid2 226 . 2 𝜓 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
61elabg 3675 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
75, 6ja 186 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1539  wcel 2107  {cab 2713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815
This theorem is referenced by:  elab3  3685  elssabg  5342  elrnmptg  5971  elrelimasn  6103  elmapg  8880  isust  24213  ellimc  25909  isismty  37809  clublem  43628
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