Theorem bj-nnflemea 37455

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.)

Assertion

Ref	Expression
bj-nnflemea	⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-nnflemea

Step	Hyp	Ref	Expression
1		bj-19.12 37389	. 2 ⊢ (∃𝑦∀𝑥𝜑 → ∀𝑥∃𝑦𝜑)
2		alim 1843	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∀𝑥∃𝑦𝜑 → ∀𝑥𝜓))
3	1, 2	syl5 35	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-nnfalt  37456