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Theorem elab2gw 3640
Description: Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2145, ax-11 2161, ax-12 2178. It also avoids a disjoint variable condition on 𝑥 and 𝐴. (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 2905 . 2 (𝐴𝐵𝐴 ∈ {𝑥𝜑})
3 elabgw.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
4 elabgw.2 . . 3 (𝑦 = 𝐴 → (𝜓𝜒))
53, 4elabgw 3639 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜒))
62, 5syl5bb 286 1 (𝐴𝑉 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  {cab 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894
This theorem is referenced by:  eldif  3918  elin  3924  elun  4100  elpwg  4514  elsng  4553  elprg  4560  eluni  4816  elintg  4859  elxpi  5554  elong  6177  isfin2  9705  iswun  10115  elnpi  10399  issal  42899
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