Theorem 3orim123da

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

	Ref	Expression
Hypotheses	3orim123da.1	$$⊢\phi \to \left(\psi \vee \theta \vee \eta \right)$$
	3orim123da.2	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
	3orim123da.3	$$⊢\left(\phi \wedge \theta \right)\to \tau$$
	3orim123da.4	$$⊢\left(\phi \wedge \eta \right)\to \zeta$$
Assertion	3orim123da	$$⊢\phi \to \left(\chi \vee \tau \vee \zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		3orim123da.1	$$⊢\phi \to \left(\psi \vee \theta \vee \eta \right)$$
2		3orim123da.2	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
3		3orim123da.3	$$⊢\left(\phi \wedge \theta \right)\to \tau$$
4		3orim123da.4	$$⊢\left(\phi \wedge \eta \right)\to \zeta$$
5	2	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
6	3	ex	$$⊢\phi \to \left(\theta \to \tau \right)$$
7	4	ex	$$⊢\phi \to \left(\eta \to \zeta \right)$$
8	5 6 7	3orim123d	$$⊢\phi \to \left(\left(\psi \vee \theta \vee \eta \right)\to \left(\chi \vee \tau \vee \zeta \right)\right)$$
9	1 8	mpd	$$⊢\phi \to \left(\chi \vee \tau \vee \zeta \right)$$