Theorem 2nexaln 1863

Description: Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2nexaln	⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Proof of Theorem 2nexaln

Step	Hyp	Ref	Expression
1		2exnaln 1862	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
2	1	bicomi 227	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ∃𝑥∃𝑦𝜑)
3	2	con1bii 359	1 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  cbvex2v  2375  cbvex2  2443  2mo  2675  bj-alcomexcom  37398  pm11.63  45206  fun2dmnopgexmpl  48159  spr0nelg  48363  pg4cyclnex  49030