$p \cup \left( { \sim p} \right)$ (Law of excluded middle)
$\sim \left( {p \cap \sim p} \right)$ (Law of contradiction)
$p \leftrightarrow \sim \left( { \sim p} \right)$ (Law of double negation). (The above 3 tautologies are widely known laws of classical logic.)
$p \cap \left( {p \to q} \right) \to q$
$\left[ {\left( {p \to q} \right) \cap \left( {p \to r} \right)} \right] \to \left[ {p \to q \cap r} \right]$
$\left[ {\left( {p \to q} \right) \cap \left( {q \to r} \right)} \right] \to \left[ {p \to r} \right]$ (principle of syllogism)
$\left[ { \sim q \to \sim p} \right] \leftrightarrow \left[ {p \to q} \right]$ (law of contrapositive)

If a condition $p \to q$ is a tautology, then we say that p implies q i.e $p \Rightarrow q$ .

Truth table

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