formal proof

a formal proof is a finite sequence of premises each of which is an axiom, an assumption, or follows from the preceding statements in the sequence by a rule of inference, which are used to prove arguments.
a proof sequence from $\text{[math]}$ to a proposition $\text{[math]}$ is one such that $\text{[math]}$ is the last proposition in the sequence which we call the conclusion. we write $\text{[math]}$

usually, on the right side of each formula we write a description of how it was derived and from which previous lines in the inference process

correct use of equivalences and implications

use of equivalences can also be done on subformulas
use of logical implications cannot be done on subformulas, only on full rows during the formal proof

an example of incorrect usage:
$\text{[math]}$