Theorem bj-elabd2ALT 34807

Description: Alternate proof of elabd2 3572 bypassing elab6g 3571 (and using sbiedvw 2100 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bj-elabd2ALT.ex	⊢ (𝜑 → 𝐴 ∈ 𝑉)
bj-elabd2ALT.eq	⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})
bj-elabd2ALT.is	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-elabd2ALT	⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))

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.

Step	Hyp	Ref	Expression
1		bj-elabd2ALT.ex	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
3		bj-elabd2ALT.eq	. . . . . 6 ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})
4	3	eqcomd 2740	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐵)
5	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → {𝑥 ∣ 𝜓} = 𝐵)
6	2, 5	eleq12d 2828	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ 𝐵))
7		eqeq1 2738	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
8	7	biimparc 483	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴)
9	8	anim2i 620	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → (𝜑 ∧ 𝑥 = 𝐴))
10	9	anassrs 471	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝐴))
11		bj-elabd2ALT.is	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
12	10, 11	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
13	12	sbiedvw 2100	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
14	6, 13	bibi12d 349	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) ↔ (𝐴 ∈ 𝐵 ↔ 𝜒)))
15		df-clab 2713	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓))
17	1, 14, 16	vtocld 3463	1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  [wsb 2070   ∈ wcel 2110  {cab 2712

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 2112  ax-9 2120  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812

This theorem is referenced by: (None)