Theorem pm11.63

Description: Theorem *11.63 in WhiteheadRussell p. 166. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.63	⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		2nexaln	⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )
2		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
3	2	2alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 → ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )
4	1 3	sylbi	⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )