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Proof by contradiction (general): assume {\displaystyle \lnot P} {\displaystyle \lnot P} and derive a contradiction.
This corresponds, in the framework of propositional logic, to the equivalence {\displaystyle P\equiv \lnot \lnot P\equiv \lnot P\to \bot } {\displaystyle P\equiv \lnot \lnot P\equiv \lnot P\to \bot }, where {\displaystyle \bot } \bot is the logical contradiction, or false value.
In the case where the statement to be proven is an implication {\displaystyle A\rightarrow B} {\displaystyle A\rightarrow B}, let us look at the differences between direct proof, proof by contrapositive, and proof by contradiction:
Direct proof: assume {\displaystyle A} A and show {\displaystyle B} B.
Proof by contrapositive: assume {\displaystyle \lnot B} {\displaystyle \lnot B} and show {\displaystyle \lnot A} {\displaystyle \lnot A}.
This corresponds to the equivalence {\displaystyle A\rightarrow B\equiv \lnot B\rightarrow \lnot A} {\displaystyle A\rightarrow B\equiv \lnot B\rightarrow \lnot A}.
Proof by contradiction: assume {\displaystyle A} A and {\displaystyle \lnot B} {\displaystyle \lnot B} and derive a contradiction.
This corresponds to the equivalences {\displaystyle A\rightarrow B\equiv \lnot \lnot (A\rightarrow B)\equiv \lnot (A\rightarrow B)\rightarrow \bot \equiv (A\land \lnot B)\rightarrow \bot } {\displaystyle A\rightarrow B\equiv \lnot \lnot (A\rightarrow B)\equiv \lnot (A\rightarrow B)\rightarrow \bot \equiv (A\land \lnot B)\rightarrow \bot }.