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Metamath Proof Explorer

Theorem elequ12

Description: An identity law for the non-logical predicate, which combines elequ1 and elequ2 . The analogous theorems for class terms are eleq1 , eleq2 , and eleq12 respectively. (Contributed by BJ, 29-Sep-2019)

		Ref	Expression
	Assertion	elequ12	$⊢ (x = y \land z = t) \to (x \in z \leftrightarrow y \in t)$

Proof

Step	Hyp	Ref	Expression
1		elequ1	$⊢ x = y \to (x \in z \leftrightarrow y \in z)$
2		elequ2	$⊢ z = t \to (y \in z \leftrightarrow y \in t)$
3	1 2	sylan9bb	$⊢ (x = y \land z = t) \to (x \in z \leftrightarrow y \in t)$