Theorem 19.21t

Description: Closed form of Theorem 19.21 of Margaris p. 90, see 19.21 . (Contributed by NM, 27-May-1997) (Revised by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 3-Jan-2018) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 3-Nov-2021)

	Ref	Expression
Assertion	19.21t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.38a	⊢ ( Ⅎ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
2		19.9t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )
3	2	imbi1d	⊢ ( Ⅎ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
4	1 3	bitr3d	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )