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Metamath Proof Explorer

Theorem bj-nnflemae

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

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

Proof

Step	Hyp	Ref	Expression
1		exim	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑦 𝜑 ) )
2		bj-19.12	⊢ ( ∃ 𝑥 ∀ 𝑦 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 ) )