Theorem imp5a

Description: An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009) (Proof shortened by Wolf Lammen, 2-Aug-2022)

	Ref	Expression
Hypothesis	imp5.1	$$⊢\phi \to \left(\psi \to \left(\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)\right)\right)$$
Assertion	imp5a	$$⊢\phi \to \left(\psi \to \left(\chi \to \left(\left(\theta \wedge \tau \right)\to \eta \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imp5.1	$$⊢\phi \to \left(\psi \to \left(\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)\right)\right)$$
2	1	imp5d	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \chi \right)\to \left(\left(\theta \wedge \tau \right)\to \eta \right)$$
3	2	exp31	$$⊢\phi \to \left(\psi \to \left(\chi \to \left(\left(\theta \wedge \tau \right)\to \eta \right)\right)\right)$$