Theorem bj-wnf2

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-wnf2	⊢ ( ∃ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		hbe1	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )
2		bj-eximcom	⊢ ( ∃ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑥 𝜓 ) )
3		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜓 → ∀ 𝑥 𝜓 )
4	1 2 3	syl56	⊢ ( ∃ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )