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Theorem 4exmid 1032
 Description: The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 878). (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 1031 . . 3 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
21biimpi 208 . 2 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
32orri 848 1 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 387   ∨ wo 833 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 199  df-an 388  df-or 834 This theorem is referenced by:  clsk1indlem3  39756
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