Theorem 19.24

Description: Theorem 19.24 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.24	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.2	⊢ ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜓 )
2	1	imim2i	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
4	2 3	sylibr	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 → 𝜓 ) )