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Theorem bj-elabd2ALT 34876
Description: Alternate proof of elabd2 3592 bypassing elab6g 3591 (and using sbiedvw 2101 instead of the 𝑥(𝑥 = 𝑦𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-elabd2ALT.ex (𝜑𝐴𝑉)
bj-elabd2ALT.eq (𝜑𝐵 = {𝑥𝜓})
bj-elabd2ALT.is ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
bj-elabd2ALT (𝜑 → (𝐴𝐵𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem bj-elabd2ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-elabd2ALT.ex . 2 (𝜑𝐴𝑉)
2 simpr 488 . . . 4 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
3 bj-elabd2ALT.eq . . . . . 6 (𝜑𝐵 = {𝑥𝜓})
43eqcomd 2744 . . . . 5 (𝜑 → {𝑥𝜓} = 𝐵)
54adantr 484 . . . 4 ((𝜑𝑦 = 𝐴) → {𝑥𝜓} = 𝐵)
62, 5eleq12d 2833 . . 3 ((𝜑𝑦 = 𝐴) → (𝑦 ∈ {𝑥𝜓} ↔ 𝐴𝐵))
7 eqeq1 2742 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
87biimparc 483 . . . . . . 7 ((𝑦 = 𝐴𝑥 = 𝑦) → 𝑥 = 𝐴)
98anim2i 620 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝐴𝑥 = 𝑦)) → (𝜑𝑥 = 𝐴))
109anassrs 471 . . . . 5 (((𝜑𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜑𝑥 = 𝐴))
11 bj-elabd2ALT.is . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1210, 11syl 17 . . . 4 (((𝜑𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜓𝜒))
1312sbiedvw 2101 . . 3 ((𝜑𝑦 = 𝐴) → ([𝑦 / 𝑥]𝜓𝜒))
146, 13bibi12d 349 . 2 ((𝜑𝑦 = 𝐴) → ((𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓) ↔ (𝐴𝐵𝜒)))
15 df-clab 2716 . . 3 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
1615a1i 11 . 2 (𝜑 → (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓))
171, 14, 16vtocld 3483 1 (𝜑 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  [wsb 2071  wcel 2111  {cab 2715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817
This theorem is referenced by: (None)
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