Theorem 4exmid

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

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

Proof

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