Theorem bj-nnflemea 34874

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 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnflemea

Step	Hyp	Ref	Expression
1		bj-19.12 34870	. 2 ⊢ (∃𝑦∀𝑥𝜑 → ∀𝑥∃𝑦𝜑)
2		alim 1814	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∀𝑥∃𝑦𝜑 → ∀𝑥𝜑))
3	1, 2	syl5 34	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  bj-nnfalt  34875