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Metamath Proof Explorer

Theorem bj-nnflemea

Description: One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	bj-nnflemea	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜑 ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-19.12	⊢ ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑦 𝜑 )
2		alim	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜑 ) → ( ∀ 𝑥 ∃ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	syl5	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜑 ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )