Theorem bj-wnfanf

Description: When ph is substituted for ps , this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnfanf	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-wnf1	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
2		bj-19.23bit	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) )
3	1 2	sylg	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) )