Theorem eleq2w2

Description: A weaker version of eleq2 (but stronger than ax-9 and elequ2 ) that uses ax-12 to avoid ax-8 and df-clel . Compare eleq2w , whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024)

	Ref	Expression
Assertion	eleq2w2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	biimpi	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
3	2	19.21bi	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )