Theorem 19.23t

Description: Closed form of Theorem 19.23 of Margaris p. 90. See 19.23 . (Contributed by NM, 7-Nov-2005) (Proof shortened by Wolf Lammen, 13-Aug-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 8-Oct-2022)

	Ref	Expression
Assertion	19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.38b	⊢ ( Ⅎ 𝑥 𝜓 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
2		19.3t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 𝜓 ↔ 𝜓 ) )
3	2	imbi2d	⊢ ( Ⅎ 𝑥 𝜓 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )
4	1 3	bitr3d	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )