Theorem elequ2g

Description: A form of elequ2 with a universal quantifier. Its converse is the axiom of extensionality ax-ext . (Contributed by BJ, 3-Oct-2019)

	Ref	Expression
Assertion	elequ2g	⊢ ( 𝑥 = 𝑦 → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		elequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
2	1	alrimiv	⊢ ( 𝑥 = 𝑦 → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )