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Theorem elab2gw 3679
Description: Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on 𝑥 and 𝐴, which is not usually significant since 𝐵 is usually a constant. (Contributed by SN, 16-May-2024.)
Hypotheses
Ref Expression
elabgw.1 (𝑥 = 𝑦 → (𝜑𝜓))
elabgw.2 (𝑦 = 𝐴 → (𝜓𝜒))
elab2gw.3 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
elab2gw (𝐴𝑉 → (𝐴𝐵𝜒))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦   𝜒,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elab2gw
StepHypRef Expression
1 elab2gw.3 . . 3 𝐵 = {𝑥𝜑}
21eleq2i 2830 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝜑})
3 elabgw.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
4 elabgw.2 . . 3 (𝑦 = 𝐴 → (𝜓𝜒))
53, 4elabgw 3678 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜒))
62, 5bitrid 283 1 (𝐴𝑉 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  {cab 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813
This theorem is referenced by:  elrab2w  3698
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