# Problem with Propositional Logic

Hi,

I've been set an assignment, part of which is to come up with a formal proof for (p $\wedge$ q) $\Rightarrow$ p. I have to show that the formula is either a tautology or contradiction, or contingent. If it is contingent, I have to show the smallest possible equivalent expression that uses only conjunction, disjunction and negation.

I'm also only allowed to use the following tautologies:

(p $\wedge$ p) $\Leftrightarrow$ p
(p $\vee$ p) $\Leftrightarrow$ p
(p $\vee \neg$p) $\Leftrightarrow$ T
((p $\vee$ q) $\vee$ r) $\Leftrightarrow$ (p $\vee$ (q $\vee$ r))
((p $\wedge$ q) $\wedge$ r) $\Leftrightarrow$ (p $\wedge$ (q $\wedge$ r))
(p $\vee$ r) $\Leftrightarrow$ (r $\vee$ p)
(p $\wedge$ r) $\Leftrightarrow$ (r $\wedge$ p)
(p $\vee$ T) $\Leftrightarrow$ T
(p $\Leftrightarrow$ p) $\Leftrightarrow$ T
$\neg$$\neg$p $\Leftrightarrow$ p
$\neg$(p $\wedge$ r) $\Leftrightarrow$ ($\neg$p $\vee$ $\neg$r)
$\neg$(p $\vee$ r) $\Leftrightarrow$ ($\neg$p $\wedge$ $\neg$r)
(p $\Rightarrow$ q) $\Leftrightarrow$ ($\neg$p $\vee$ r)

My first thought was to use the last tautology in this way:
(p $\wedge$ q) $\Rightarrow$ p $\Leftrightarrow$ $\neg$(p $\wedge$ q) $\vee$ p
Firstly, I'm not entirely sure I can even use it in that way, and even if I can, I'm not sure what to do next. I can see that $\neg$(p $\wedge$ q) $\vee$ p always evaluates to true, but I've spend a good few hours on this and still can't see how I can use the above tautologies to prove it.

Any help with this would be greatly appreciated :)

EDIT: Nevermind, finally got it :)

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