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Could you clarify the relationship between proofs that use ≤ and those that use <?

For example, if it's already proven that "abs(b) ≤ a if and only if -a≤ b≤a" can we say this implies that "abs(b) < a if and only if -a< b<a"? It seems that since the first statement holds for all abs(b) "less than or equal to" a, its application could narrowed to the case where abs(b) was simply "less than" a.

Thanks!