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Theorem elabgw 3639
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. This is to elabg 3641 what sbievw2 2107 is to sbievw 2103. (Contributed by SN, 20-Apr-2024.)
Hypotheses
Ref Expression
elabgw.1 (𝑥 = 𝑦 → (𝜑𝜓))
elabgw.2 (𝑦 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
elabgw (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜒))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦   𝜒,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabgw
StepHypRef Expression
1 eleq1 2901 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
2 elabgw.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 df-clab 2801 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 elabgw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
54sbievw 2103 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
63, 5bitri 278 . 2 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
71, 2, 6vtoclbg 3544 1 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  [wsb 2069  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:  elab2gw  3640  elgch  10033
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