Theorem bj-nnflemae 34946

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 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))

Proof of Theorem bj-nnflemae

Step	Hyp	Ref	Expression
1		exim 1836	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜑))
2		bj-19.12 34943	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  bj-nnfext  34949