We can rewrite formulas, with the aim of better understanding the meaning of the logical ‘bones’ (=form) of the statement, and to produce provable formulas, or proofs.
If the rewritten formula has the same meaning as its original formula, both formulas are equivalent (≡). If instead we place a biconditional (↔) between the rewritten and its original formula, and we calculate under this operand, then we get a tautology.
1. Same meaning
Rewriting formulas with operands ˅, ˄, ~
If we use the operands ˅, ˄, ~, which together form a functionally complete set, then we can rewrite these formulas using the following rules (for statements with 2 variables and 1 operand):
Rewriting the implication
Expressions that are often used to rewrite the implication.
2. Different meaning
Expressions opposing/contrary to the implication
expression 
form 
description of the operation 
outcome^{1)} 
diagram 
inverse 
~p→~q 
negation of both statements 
1101 

converse 
q→p 
reversal of both statements 
1101 
Negating the implication
expression 
form 
description of the operation 
outcome^{1)} 
diagram 
negation 
p˄~q 
contradicts the implication 
0100 
^{1)} For spatial efficiency, instead of vertically, I noted the formula’s outcome in the truth table columns horizontally and between .