Theorem bj-nnftht

Description: A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp (modal T), as in bj-nnfbi . (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnftht	⊢ ( ( 𝜑 ∧ ∀ 𝑥 𝜑 ) → Ⅎ' 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
2		ax-1	⊢ ( ∀ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	anim12i	⊢ ( ( 𝜑 ∧ ∀ 𝑥 𝜑 ) → ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) )
4		df-bj-nnf	⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) )
5	3 4	sylibr	⊢ ( ( 𝜑 ∧ ∀ 𝑥 𝜑 ) → Ⅎ' 𝑥 𝜑 )