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Metamath Proof Explorer

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 e. A <-> x e. B ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	\|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
2	1	biimpi	\|- ( A = B -> A. x ( x e. A <-> x e. B ) )
3	2	19.21bi	\|- ( A = B -> ( x e. A <-> x e. B ) )