Theorem 2nexaln 1832

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 1831	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
2	1	bicomi 223	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ∃𝑥∃𝑦𝜑)
3	2	con1bii 357	1 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  cbvex2v  2342  cbvex2  2412  2mo  2650  bj-alcomexcom  34862  pm11.63  42013  fun2dmnopgexmpl  44776  spr0nelg  44928