Equivalences:

In this following miscellaneous equivalence rules are often useful during rewriting sessions. So there the first two allow us to completely get rid of implication and equivalence connectives from our sentences.

However there the "False" symbol stands for the proposition that is always false: does not matter that what truth values you give to other propositions in the sentence and this one will always be false. Same like this the "True" symbol stands for the proposition that is always true. According to the first-order logic we can treat them as special predicates with the same properties.