In
mathematics and
logic, a
direct proof is a way of showing the
truth or falsehood of a given statement by a straightforward combination of established facts, usually
axioms, existing
lemmas and
theorems, without making any further assumptions. In order to directly prove a
conditional statement of the form "If
p, then
q", it suffices to consider the situations in which the statement
p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably
first-order logic, employing the quantifiers
for all and
there exists. Common proof rules used are
modus ponens and
universal instantiation.