Theorem bj-3exbi

Description: Closed form of 3exbii . (Contributed by BJ, 6-May-2019)

	Ref	Expression
Assertion	bj-3exbi	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		exbi	⊢ ( ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 ) )
2	1	2alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ∀ 𝑦 ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 ) )
3		bj-2exbi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 ) )