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Theorem elab3g 3326
 Description: Membership in a class abstraction, with a weaker antecedent than elabg 3320. (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 nfcv 2751 . 2 𝑥𝐴
2 nfv 1830 . 2 𝑥𝜓
3 elab3g.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elab3gf 3325 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175 This theorem is referenced by:  elab3  3327  elssabg  4746  elrnmptg  5296  elrelimasn  5408  elmapg  7757  isust  21817  ellimc  23443  isismty  32770  clublem  36936
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