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Metamath Proof Explorer

Theorem elsb1

Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 for substitution for the second argument. (Contributed by NM, 7-Nov-2006) (Proof shortened by Andrew Salmon, 14-Jun-2011) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023)

		Ref	Expression
	Assertion	elsb1	$⊢ [y/ x] x \in z \leftrightarrow y \in z$

Proof

Step	Hyp	Ref	Expression
1		elequ1	$⊢ x = w \to (x \in z \leftrightarrow w \in z)$
2		elequ1	$⊢ w = y \to (w \in z \leftrightarrow y \in z)$
3	1 2	sbievw2	$⊢ [y/ x] x \in z \leftrightarrow y \in z$