Theorem 3orim123da

Description: Disjoin antecedents and consequents of three premises. (Contributed by Thierry Arnoux, 13-Jul-2026)

	Ref	Expression
Hypotheses	3orim123da.1	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) )
	3orim123da.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	3orim123da.3	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
	3orim123da.4	⊢ ( ( 𝜑 ∧ 𝜂 ) → 𝜁 )
Assertion	3orim123da	⊢ ( 𝜑 → ( 𝜒 ∨ 𝜏 ∨ 𝜁 ) )

Proof

Step	Hyp	Ref	Expression
1		3orim123da.1	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) )
2		3orim123da.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3		3orim123da.3	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
4		3orim123da.4	⊢ ( ( 𝜑 ∧ 𝜂 ) → 𝜁 )
5	2	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
6	3	ex	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
7	4	ex	⊢ ( 𝜑 → ( 𝜂 → 𝜁 ) )
8	5 6 7	3orim123d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) → ( 𝜒 ∨ 𝜏 ∨ 𝜁 ) ) )
9	1 8	mpd	⊢ ( 𝜑 → ( 𝜒 ∨ 𝜏 ∨ 𝜁 ) )