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Theorem 2exnaln 1872
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 1824 . 2 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑)
2 alnex 1825 . . 3 (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑)
32albii 1863 . 2 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑)
41, 3xchbinxr 327 1 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wal 1599  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853
This theorem depends on definitions:  df-bi 199  df-ex 1824
This theorem is referenced by:  2nexaln  1873  excom  2155  opab0  5244  bj-modal4e  33293  bj-cbvex2v  33323
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