Rule of Material Implication

Theorem
The rule of material implication is a valid deduction sequent in propositional logic:


 * $$p \implies q \dashv \vdash \neg p \or q$$

That is:
 * If one statement implies a second, then either the first is false or the second is true.

Alternative Rendition
It can alternatively be rendered as:
 * $$\vdash \left({p \implies q}\right) \iff \left({\neg p \or q}\right)$$

This can be seen to be logically equivalent to the form above.

Proof by Natural Deduction
By the tableau method:

Note that the latter proof requires the Law of Excluded Middle.

Proof by Truth Table
As can be seen by inspection, the truth values under the main connectives match for all models.

$$\begin{array}{|ccc||cccc|} \hline p & \implies & q & \neg & p & \or & q \\ \hline F & T & F & T & F & T & F \\ F & T & T & T & F & T & T \\ T & F & F & F & T & F & F \\ T & T & T & F & T & T & T \\ \hline \end{array}$$

Also see
The following are related argument forms:
 * Modus Ponendo Tollens
 * Modus Tollendo Tollens