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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 → ¬ 𝜃 )
	Assertion	3bior1fand	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∨ 𝜒 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	⊢ ( 𝜑 → ¬ 𝜃 )
2	1	intnanrd	⊢ ( 𝜑 → ¬ ( 𝜃 ∧ 𝜏 ) )
3	2	3bior1fd	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∨ 𝜒 ∨ 𝜓 ) ) )