Theorem axextb

Description: A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext and df-cleq . (Contributed by NM, 14-Nov-2008)

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

Proof

Step	Hyp	Ref	Expression
1		elequ2g	⊢ ( 𝑥 = 𝑦 → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
2		axextg	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )
3	1 2	impbii	⊢ ( 𝑥 = 𝑦 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )