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Theorem 2nalexn 1795
 Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2nalexn (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)

Proof of Theorem 2nalexn
StepHypRef Expression
1 df-ex 1745 . . 3 (∃𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
2 alex 1793 . . . 4 (∀𝑦𝜑 ↔ ¬ ∃𝑦 ¬ 𝜑)
32albii 1787 . . 3 (∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∃𝑦 ¬ 𝜑)
41, 3xchbinxr 324 . 2 (∃𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑥𝑦𝜑)
54bicomi 214 1 (¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  spc2gv  3327  hashfun  13262  spc2d  29441  pm11.52  38903  2exanali  38904
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