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Metamath Proof Explorer

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	$⊢ (((φ \land ψ) \lor (\neg φ \land \neg ψ)) \lor ((φ \land \neg ψ) \lor (ψ \land \neg φ)))$

Proof

Step	Hyp	Ref	Expression
1		pm5.24	$⊢ \neg ((φ \land ψ) \lor (\neg φ \land \neg ψ)) \leftrightarrow ((φ \land \neg ψ) \lor (ψ \land \neg φ))$
2	1	biimpi	$⊢ \neg ((φ \land ψ) \lor (\neg φ \land \neg ψ)) \to ((φ \land \neg ψ) \lor (ψ \land \neg φ))$
3	2	orri	$⊢ (((φ \land ψ) \lor (\neg φ \land \neg ψ)) \lor ((φ \land \neg ψ) \lor (ψ \land \neg φ)))$