Theorem 3imp3i2an

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017) (Proof shortened by Wolf Lammen, 13-Apr-2022)

	Ref	Expression
Hypotheses	3imp3i2an.1	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$
	3imp3i2an.2	$$⊢\left(\phi \wedge \chi \right)\to \tau$$
	3imp3i2an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
Assertion	3imp3i2an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		3imp3i2an.1	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$
2		3imp3i2an.2	$$⊢\left(\phi \wedge \chi \right)\to \tau$$
3		3imp3i2an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
4	2	3adant2	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \tau$$
5	1 4 3	syl2anc	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \eta$$