Theorem alexnim 1648

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 1646	. . 3 ⊢ (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
2	1	alimi 1455	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3		alnex 1499	. 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
4	2, 3	sylib 122	1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461

This theorem is referenced by:  nalset  4132  bj-nalset  14507