Theorem eleq2w2ALT 37740

Description: Alternate proof of eleq2w2 2761 and special instance of eleq2 2854. (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 2758	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2	1	biimpi 219	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
3		eleq1w 2848	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		eleq1w 2848	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
5	3, 4	bibi12d 348	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
6	5	spvv 2021	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
7	2, 6	syl 18	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-clel 2840

This theorem is used by: (None)