Theorem 19.25

Description: Theorem 19.25 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.25	⊢ ( ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∃ 𝑦 ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2	1	biimpi	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	2	aleximi	⊢ ( ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∃ 𝑦 ∃ 𝑥 𝜓 ) )