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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 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 )

Proof

Step	Hyp	Ref	Expression
1		elequ1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧 ) )
2		elequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 )