Theorem 4exmid 1067

Description: The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 908). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)

Assertion

Ref	Expression
4exmid	⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Proof of Theorem 4exmid

Step	Hyp	Ref	Expression
1		pm5.24 1066	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
2	1	biimpi 219	. 2 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
3	2	orri 876	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  clsk1indlem3  44810