Theorem 3jaaoOLD

Description: Obsolete version of 3jaao as of 16-Jun-2026. 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 modification is discouraged.) (New usage is discouraged.)

	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	3jaaoOLD	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\left(\psi \vee \tau \vee \zeta \right)\to \chi \right)$$

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	1	3ad2ant1	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\psi \to \chi \right)$$
5	2	3ad2ant2	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\tau \to \chi \right)$$
6	3	3ad2ant3	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\zeta \to \chi \right)$$
7	4 5 6	3jaod	$$⊢\left(\phi \wedge \theta \wedge \eta \right)\to \left(\left(\psi \vee \tau \vee \zeta \right)\to \chi \right)$$