Theorem alexnim 1536

Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
alexnim	⊢ (∀x∃y ¬ φ → ¬ ∃x∀yφ)

Proof of Theorem alexnim

Step	Hyp	Ref	Expression
1		exnalim 1534	. . 3 ⊢ (∃y ¬ φ → ¬ ∀yφ)
2	1	alimi 1341	. 2 ⊢ (∀x∃y ¬ φ → ∀x ¬ ∀yφ)
3		alnex 1385	. 2 ⊢ (∀x ¬ ∀yφ ↔ ¬ ∃x∀yφ)
4	2, 3	sylib 127	1 ⊢ (∀x∃y ¬ φ → ¬ ∃x∀yφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347

This theorem is referenced by:  nalset  3878  bj-nalset  9350