Theorem elsb2

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023)

	Ref	Expression
Assertion	elsb2	$$⊢\left[y/x\right]z\in x\leftrightarrow z\in y$$

Proof

Step	Hyp	Ref	Expression
1		elequ2	$$⊢x=w\to \left(z\in x\leftrightarrow z\in w\right)$$
2		elequ2	$$⊢w=y\to \left(z\in w\leftrightarrow z\in y\right)$$
3	1 2	sbievw2	$$⊢\left[y/x\right]z\in x\leftrightarrow z\in y$$