Theorem 2nexaln 1930

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 1929	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
2	1	bicomi 216	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ∃𝑥∃𝑦𝜑)
3	2	con1bii 348	1 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1656  ∃wex 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910

This theorem depends on definitions:  df-bi 199  df-ex 1881

This theorem is referenced by:  cbvex2  2431  2mo  2731  bj-alcomexcom  33205  pm11.63  39435  fun2dmnopgexmpl  42187  spr0nelg  42573