Theorem bj-19.23t

Description: Statement 19.23t proved from modalK (obsoleting 19.23v ). (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.23t	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-exlim	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜓 ) ) )
2		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜓 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
3	2	imim2d	⊢ ( Ⅎ' 𝑥 𝜓 → ( ( ∃ 𝑥 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )
4		19.38	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
5	3 4	syl6	⊢ ( Ⅎ' 𝑥 𝜓 → ( ( ∃ 𝑥 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
6	1 5	impbid	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )