Consider the following premises:

If it is icy or raining, then it is wet outside.

If it is raining, then it is dark.

You do not need a jacket.

If it is wet outside or snowing, then you need a jacket.

$($a$)$ Translate these premises into propositional logic. Use only the following atomic propositions: i: icy, r: raining, w: wet, s: snowing, d: dark, j: jacket.

$($b$)$ Use inference to show that the conclusion $"$ noticyvvdark $",$ in other words, notivvd, can be drawn from the premises above. Use the following template. Label each step below with the appropriate rule or law and specify to which line$($s$)$ it is applied to$.$ Use as many rows as needed. Don't skip steps, apply one law or one rule of inference at a time.

premise $1$

premise $2$

$.$premise $3$

premise $4$

from using

from using

$7$