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Theorem 4exmid 1039
 Description: The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 430). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)
Assertion
Ref Expression
4exmid (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Proof of Theorem 4exmid
StepHypRef Expression
1 pm5.24 1038 . . 3 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
21biimpi 206 . 2 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
32orri 390 1 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385 This theorem is referenced by:  clsk1indlem3  38861
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