Theorem alexnim 1585

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 1583	. . 3 ⊢ (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
2	1	alimi 1390	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3		alnex 1434	. 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
4	2, 3	sylib 121	1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1288  ∃wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396

This theorem is referenced by:  nalset  3975  bj-nalset  12052