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 ∧ 𝜓 ) → 𝜒 )
	mpjao3dan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
	mpjao3dan.3	⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜒 )
	mpjao3dan.4	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ∨ 𝜏 ) )
Assertion	mpjao3dan	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		mpjao3dan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		mpjao3dan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
3		mpjao3dan.3	⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜒 )
4		mpjao3dan.4	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ∨ 𝜏 ) )
5	1 2 3	3jaodan	⊢ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜃 ∨ 𝜏 ) ) → 𝜒 )
6	4 5	mpdan	⊢ ( 𝜑 → 𝜒 )