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Metamath Proof Explorer

Theorem bj-wnfenf

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

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

Proof

Step	Hyp	Ref	Expression
1		bj-wnf1	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
2		bj-19.21bit	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜓 ) )
3	1 2	sylg	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜓 ) )