Theorem bj-19.21t

Description: Statement 19.21t proved from modalK (obsoleting 19.21v ). (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.21t	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-alrim	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
2		bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
3	2	imim1d	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )
4		19.38	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
5	3 4	syl6	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
6	1 5	impbid	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )