Theorem 3jcad

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005)

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

Proof

Step	Hyp	Ref	Expression
1		3jcad.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		3jcad.2	$$⊢\phi \to \left(\psi \to \theta \right)$$
3		3jcad.3	$$⊢\phi \to \left(\psi \to \tau \right)$$
4	1	imp	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
5	2	imp	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
6	3	imp	$$⊢\left(\phi \wedge \psi \right)\to \tau$$
7	4 5 6	3jca	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \wedge \theta \wedge \tau \right)$$
8	7	ex	$$⊢\phi \to \left(\psi \to \left(\chi \wedge \theta \wedge \tau \right)\right)$$