Theorem 3anim123i

Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994)

	Ref	Expression
Hypotheses	3anim123i.1	$$⊢\phi \to \psi$$
	3anim123i.2	$$⊢\chi \to \theta$$
	3anim123i.3	$$⊢\tau \to \eta$$
Assertion	3anim123i	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \left(\psi \wedge \theta \wedge \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		3anim123i.1	$$⊢\phi \to \psi$$
2		3anim123i.2	$$⊢\chi \to \theta$$
3		3anim123i.3	$$⊢\tau \to \eta$$
4	1	3ad2ant1	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \psi$$
5	2	3ad2ant2	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \theta$$
6	3	3ad2ant3	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \eta$$
7	4 5 6	3jca	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \left(\psi \wedge \theta \wedge \eta \right)$$