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	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
2	1	biimpi	$$⊢A=B\to \forall x\left(x\in A\leftrightarrow x\in B\right)$$
3	2	19.21bi	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$