Theorem bj-nnflemea 37392

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 37326	. 2 ⊢ (∃𝑦∀𝑥𝜑 → ∀𝑥∃𝑦𝜑)
2		alim 1840	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∀𝑥∃𝑦𝜑 → ∀𝑥𝜓))
3	1, 2	syl5 35	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  bj-nnfalt  37393