Theorem 2exbi

Description: Theorem *11.341 in WhiteheadRussell p. 162. Theorem 19.18 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2exbi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

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