Theorem bj-nnfe

Description: Nonfreeness implies the equivalent of ax5e . (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-bj-nnf	⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) )
2	1	simplbi	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )