Destructive dilemma

$${\displaystyle \frac{P\to Q,R\to S,\mathrm{\neg }Q\vee \mathrm{\neg }S}{\therefore \mathrm{\neg }P\vee \mathrm{\neg }R}}$$

where the rule is that wherever instances of "${\displaystyle P\to Q}$", "${\displaystyle R\to S}$", and "${\displaystyle \mathrm{\neg }Q\vee \mathrm{\neg }S}$" appear on lines of a proof, "${\displaystyle \mathrm{\neg }P\vee \mathrm{\neg }R}$" can be placed on a subsequent line.

Formal notation

The destructive dilemma rule may be written in sequent notation:

$${\displaystyle (P\to Q),(R\to S),(\mathrm{\neg }Q\vee \mathrm{\neg }S)⊢(\mathrm{\neg }P\vee \mathrm{\neg }R)}$$

where ${\displaystyle ⊢}$ is a metalogical symbol meaning that ${\displaystyle \mathrm{\neg }P\vee \mathrm{\neg }R}$ is a syntactic consequence of ${\displaystyle P\to Q}$, ${\displaystyle R\to S}$, and ${\displaystyle \mathrm{\neg }Q\vee \mathrm{\neg }S}$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

$${\displaystyle (((P\to Q)\wedge (R\to S))\wedge (\mathrm{\neg }Q\vee \mathrm{\neg }S))\to (\mathrm{\neg }P\vee \mathrm{\neg }R)}$$

where ${\displaystyle P}$, ${\displaystyle Q}$, ${\displaystyle R}$ and ${\displaystyle S}$ are propositions expressed in some formal system.