I am relatively new to mathematics, but in the span of a few months, I have knocked out an informal proof/logic book (Solow), an abstract algebra book (Pinter), and am now reading through a real analysis book (Tao). I have developed the mechanical habits of how/when to write a particular styled proof...e.g. getting a sense of when proof by contradiction is easier...a growing familiarity with using contrapositives...etc. However, it occurred to me after posing this question here (When is proving the truth of a biconditional statement "the same" as proving that two propositions are logically equivalent?), that I still have no clue what exactly I am really doing when constructing a formal proof.
I think my major issue is that I am conflating the toolkit (and terminology) used in propositional logic with the toolkit (and terminology) used in predicate logic. The issue that I am having is differentiating between logical equivalency and logical entailment. When I am asked to prove some claim, for example, "If $A\subseteq B$, then $A \cup C \subseteq B \cup C$", then I have proven the statement if/when I show that the truth of the antecedent can be used to produce the truth of the consequent. This, I believe, is the premise of logical entailment.
However, let's say that I was having difficulties finding a direct proof of this. So, noting the logical equivalency between implication and contrapositive, I used a contrapositive-styled proof to demonstrate that I can use the negation of the consequent to produce the negation of the antecedent. ...Basically, it seems as though I am using logical equivalency and carry out a logical entailment.
This issue is exacerbated when I am asked in a predicate logic setting to prove that two statements are logically equivalent. In this context, I believe "logical equivalence" is better described as "reciprocal logical entailment", but I am not positive. Any help would be greatly appreciated!