Theorem 19.36vv

Description: Theorem *11.43 in WhiteheadRussell p. 163. Theorem 19.36 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	19.36vv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.36v	⊢ ( ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑦 𝜑 → 𝜓 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∃ 𝑥 ( ∀ 𝑦 𝜑 → 𝜓 ) )
3		19.36v	⊢ ( ∃ 𝑥 ( ∀ 𝑦 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜓 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜓 ) )