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Metamath Proof Explorer

Theorem 3bior1fand

Description: A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017)

		Ref	Expression
	Hypothesis	3biorfd.1	$⊢ φ \to \neg θ$
	Assertion	3bior1fand	$⊢ φ \to ((χ \lor ψ) \leftrightarrow ((θ \land τ) \lor χ \lor ψ))$

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	$⊢ φ \to \neg θ$
2	1	intnanrd	$⊢ φ \to \neg (θ \land τ)$
3	2	3bior1fd	$⊢ φ \to ((χ \lor ψ) \leftrightarrow ((θ \land τ) \lor χ \lor ψ))$