Theorem alexnim 1582

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 1580	. . 3 ⊢ (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
2	1	alimi 1387	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3		alnex 1431	. 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
4	2, 3	sylib 120	1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1285  ∃wex 1424

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393

This theorem is referenced by:  nalset  3937  bj-nalset  11143