Theorem elequ1

Description: An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993)

	Ref	Expression
Assertion	elequ1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		ax8	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) )
2		ax8	⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧 ) )
3	2	equcoms	⊢ ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧 ) )
4	1 3	impbid	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )