Theorem bj-nnfea

Description: Nonfreeness implies the equivalent of ax5ea . (Contributed by BJ, 28-Jul-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
2		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	syld	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )