Theorem bj-nnflemae 37441

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.)

Assertion

Ref	Expression
bj-nnflemae	⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))

Proof of Theorem bj-nnflemae

Step	Hyp	Ref	Expression
1		exim 1863	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜓))
2		bj-19.12 37376	. 2 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓)
3	1, 2	syl6 36	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  bj-nnfext  37444