Why logical equivalences are important
The replacement of a (sub)statement with a logically equivalent
statement is a powerful tool in mathematical reasoning. The
result of such a replacement will always have the same truth
value as (will be logically equivalent to) the original statement.
E.g., ?(? p) ? q ? p ? q, because ?(? p) ? p.
Also, using a chain of this kind of replacement can sometimes be
the only way to establish that two propositions are logically
equivalent. For the truth table for a compound proposition with 100
variables would have 2 rows!