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Metamath Proof Explorer

Theorem mpjao3dan

Description: Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018) (Proof shortened by Wolf Lammen, 20-Apr-2024)

	Ref	Expression
Hypotheses	mpjao3dan.1	$⊢ (φ \land ψ) \to χ$
	mpjao3dan.2	$⊢ (φ \land θ) \to χ$
	mpjao3dan.3	$⊢ (φ \land τ) \to χ$
	mpjao3dan.4	$⊢ φ \to (ψ \lor θ \lor τ)$
Assertion	mpjao3dan	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		mpjao3dan.1	$⊢ (φ \land ψ) \to χ$
2		mpjao3dan.2	$⊢ (φ \land θ) \to χ$
3		mpjao3dan.3	$⊢ (φ \land τ) \to χ$
4		mpjao3dan.4	$⊢ φ \to (ψ \lor θ \lor τ)$
5	1 2 3	3jaodan	$⊢ (φ \land (ψ \lor θ \lor τ)) \to χ$
6	4 5	mpdan	$⊢ φ \to χ$