Theorem 3jaao

Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Andrew Salmon, 13-May-2011) (Proof shortened by Garrett Katz, 16-Jun-2026)

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

Proof

Step	Hyp	Ref	Expression
1		3jaao.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3jaao.2	$$⊢\theta \to \left(\tau \to \chi \right)$$
3		3jaao.3	$$⊢\eta \to \left(\zeta \to \chi \right)$$
4		3jao	$$⊢\left(\left(\psi \to \chi \right)\wedge \left(\tau \to \chi \right)\wedge \left(\zeta \to \chi \right)\right)\to \left(\left(\psi \vee \tau \vee \zeta \right)\to \chi \right)$$
5	1 2 3 4	syl3an	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\left(\psi \vee \tau \vee \zeta \right)\to \chi \right)$$