Theorem jca3

Description: Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010)

	Ref	Expression
Hypotheses	jca3.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	jca3.2	$$⊢\theta \to \tau$$
Assertion	jca3	$$⊢\phi \to \left(\psi \to \left(\theta \to \left(\chi \wedge \tau \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		jca3.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		jca3.2	$$⊢\theta \to \tau$$
3	1	imp	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
4	3	a1d	$$⊢\left(\phi \wedge \psi \right)\to \left(\theta \to \chi \right)$$
5	4 2	jca2	$$⊢\left(\phi \wedge \psi \right)\to \left(\theta \to \left(\chi \wedge \tau \right)\right)$$
6	5	ex	$$⊢\phi \to \left(\psi \to \left(\theta \to \left(\chi \wedge \tau \right)\right)\right)$$