Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  4exmid Structured version   Visualization version   GIF version

Theorem 4exmid 1047
 Description: The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 892). (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 1046 . . 3 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
21biimpi 219 . 2 (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
32orri 859 1 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by:  clsk1indlem3  41164
 Copyright terms: Public domain W3C validator