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 → ∀ 𝑦𝑥 𝜑 ) )

Proof

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