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	⊢ ( [ 𝑦 / 𝑥 ] 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		elequ2	⊢ ( 𝑥 = 𝑤 → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤 ) )
2		elequ2	⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑥 ] 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 )