Theorem elequ2

Description: An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	elequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		ax9	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) )
2		ax9	⊢ ( 𝑦 = 𝑥 → ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥 ) )
3	2	equcoms	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥 ) )
4	1 3	impbid	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )