Theorem 3orim123d

Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997)

	Ref	Expression
Hypotheses	3anim123d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	3anim123d.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
	3anim123d.3	$$⊢\phi \to \left(\eta \to \zeta \right)$$
Assertion	3orim123d	$$⊢\phi \to \left(\left(\psi \vee \theta \vee \eta \right)\to \left(\chi \vee \tau \vee \zeta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3anim123d.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3anim123d.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
3		3anim123d.3	$$⊢\phi \to \left(\eta \to \zeta \right)$$
4	1 2	orim12d	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$
5	4 3	orim12d	$$⊢\phi \to \left(\left(\left(\psi \vee \theta \right)\vee \eta \right)\to \left(\left(\chi \vee \tau \right)\vee \zeta \right)\right)$$
6		df-3or	$$⊢\left(\psi \vee \theta \vee \eta \right)\leftrightarrow \left(\left(\psi \vee \theta \right)\vee \eta \right)$$
7		df-3or	$$⊢\left(\chi \vee \tau \vee \zeta \right)\leftrightarrow \left(\left(\chi \vee \tau \right)\vee \zeta \right)$$
8	5 6 7	3imtr4g	$$⊢\phi \to \left(\left(\psi \vee \theta \vee \eta \right)\to \left(\chi \vee \tau \vee \zeta \right)\right)$$