In the if-clauseA group of related words containing a subject and a verb. of the first conditional we can use should (to express that the condition is less likely), can (permission or ability), will or would (willingness, insistence and refusal), must (obligation) or may (permission).

If you can speak English, you may apply for this position.
If you must cry, cry.
If you should see Kate, could you ask her about my CD?

Politeness

Will and would in the if-clause can express willingness.

If you will/would wait here (, I’ll see if the boss is in). (request)
If you’d just fill in this form. (polite request, e.g. in a hotel)

In these two cases inversionA structure in which a verb or an auxiliary precedes the subject. E.g.: "Here comes the sun." is possible:

Will you wait here (, I'll see if the boss is in)?
Would you fill in this form?

Since in these cases the result is obvious or not important, the main clauseA clause that could be a sentence in itself. E.g.: I wouldn’t go there if I were you. ("I wouldn't go there") When I arrived he was surfing. ("he was surfing") is often omitted.

Insistence/willingness or refusal/unwillingness

If he will walk around in a T-shirt at this time of the year, he will surely catch a cold. (= If he insists on walking...)
If she won’t greet me, I won’t greet her either. (= If she refuses to greet me...)

Note that in these two sentences the auxiliariesA helping verb used with main verbs to express mood, voice and tense. E.g.: I can't stand when I am disturbed. ("can", "am") will and won’t in the if-clause are always stressed in speech and cannot be contracted with the subjectThe person or thing that performs the action of a verb or that is described by a verb. E.g.: Andy has never been to Brunei. ("Andy"):

If he’ll walk around in a T-shirt in the winter, no wonder he catches colds.

Modals in first conditional
Modals in first conditional Modals in the main clause of first conditional sentences