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Theorem 2exnaln 1903
Description: Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2exnaln (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Proof of Theorem 2exnaln
StepHypRef Expression
1 df-ex 1852 . 2 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑)
2 alnex 1853 . . 3 (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑)
32albii 1894 . 2 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑)
41, 3xchbinxr 324 1 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884
This theorem depends on definitions:  df-bi 197  df-ex 1852
This theorem is referenced by:  2nexaln  1904  excom  2197  opab0  5140  bj-modal4e  33036  bj-cbvex2v  33068
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