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Theorem elabd3 3654
Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab 3661. (Contributed by Gino Giotto, 12-Oct-2024.)
Hypotheses
Ref Expression
elabd3.ex (𝜑𝐴𝑉)
elabd3.is ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
elabd3 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem elabd3
StepHypRef Expression
1 elabd3.ex . 2 (𝜑𝐴𝑉)
2 eqidd 2725 . 2 (𝜑 → {𝑥𝜓} = {𝑥𝜓})
3 elabd3.is . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
41, 2, 3elabd2 3653 1 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802
This theorem is referenced by:  sbcied  3815
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