Theorem bj-nnfeai

Description: Nonfreeness implies the equivalent of ax5ea , inference form. (Contributed by BJ, 22-Sep-2024)

	Ref	Expression
Hypothesis	bj-nnfeai.1	⊢ Ⅎ' 𝑥 𝜑
Assertion	bj-nnfeai	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfeai.1	⊢ Ⅎ' 𝑥 𝜑
2		bj-nnfea	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	ax-mp	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )