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

Proof

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