Theorem pm11.61

Description: Theorem *11.61 in WhiteheadRussell p. 166. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.61	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.12	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) )
2		19.37v	⊢ ( ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑦 𝜓 ) )
3	2	biimpi	⊢ ( ∃ 𝑦 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑦 𝜓 ) )
4	3	alimi	⊢ ( ∀ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∃ 𝑦 𝜓 ) )
5	1 4	syl	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∃ 𝑦 𝜓 ) )