Theorem eleq2w2ALT 37013

Description: Alternate proof of eleq2w2 2736 and special instance of eleq2 2833. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eleq2w2ALT	⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Proof of Theorem eleq2w2ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2733	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2	1	biimpi 216	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
3		eleq1w 2827	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		eleq1w 2827	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
5	3, 4	bibi12d 345	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
6	5	spvv 1996	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
7	2, 6	syl 17	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-clel 2819

This theorem is referenced by: (None)