Theorem anim12dan

Description: Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	anim12dan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
	anim12dan.2	$$⊢\left(\phi \wedge \theta \right)\to \tau$$
Assertion	anim12dan	$$⊢\left(\phi \wedge \left(\psi \wedge \theta \right)\right)\to \left(\chi \wedge \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		anim12dan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		anim12dan.2	$$⊢\left(\phi \wedge \theta \right)\to \tau$$
3	1	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
4	2	ex	$$⊢\phi \to \left(\theta \to \tau \right)$$
5	3 4	anim12d	$$⊢\phi \to \left(\left(\psi \wedge \theta \right)\to \left(\chi \wedge \tau \right)\right)$$
6	5	imp	$$⊢\left(\phi \wedge \left(\psi \wedge \theta \right)\right)\to \left(\chi \wedge \tau \right)$$