Theorem bj-wnf1

Description: When ph is substituted for ps , this is the first half of nonfreness ( . -> A. ) of the weak form of nonfreeness ( E. -> A. ) . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnf1	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-modal4e	⊢ ( ∃ 𝑥 ∃ 𝑥 𝜑 → ∃ 𝑥 𝜑 )
2		hba1	⊢ ( ∀ 𝑥 𝜓 → ∀ 𝑥 ∀ 𝑥 𝜓 )
3	1 2	imim12i	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 ∃ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜓 ) )
4		19.38	⊢ ( ( ∃ 𝑥 ∃ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
5	3 4	syl	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )