Where one thing implies another

Solution 1:

Entails is one possibility for the verb, but it is formal:

entail [tr.v.]

To have, impose, or require as a necessary accompaniment or consequence:

• The investment entailed a high risk.

[AHD]

entail

To entail is to [necessarily] involve. A job at a movie theater might entail sweeping popcorn off the floor, probably because

• watching a movie entails eating popcorn in the dark.

[Vocabulary.com]

...............

Another even more formal term for the strict mathematical sense of imply (A → B; if A is true, so is B) is predicate:

predicate – (logic) what is predicated of the subject of a proposition; the second term in a proposition is predicated of the first term by means of the copula:

• `Socrates is a man' predicates manhood of Socrates.

[WordNet 3.0]

..............

However, possibly the most idiomatic way to say this sort of thing in less formal registers is:

• 'Arrogance is synonymous with insecurity.'
• 'Strength is synonymous with failure.' etc.

If you say that one thing is synonymous with another, you mean that the two things are very closely associated with each other so that one suggests the other or one cannot exist without the other.

[CollinsCoBuild]

• 'Capitalism is synonymous with greed.'

[PoetChristopher] (obviously voicing an opinion)

This is obviously a stretched definition of 'synonymous'.

Solution 2:

A few words and phrases come to mind.

• Arrogance stems from insecurity. (a cause/effect relatioship)
• Arrogance and insecurity usually go together. (not necessarily cause/effect)
• Arrogance is usually attached to insecurity. (neutral)
• Arrogance dwells in insecurity.
• Arrogance lies in insecurity.

As for the "strength/failure" example, I can't see how you could have them related, unless you admit that "stength prevents failure.