# Prove that $((p\lor q)\land(p\lor(\lnot q)))\rightarrow p$ is a tautology

Prove that $((p\lor q)\land(p\lor(\lnot q)))\rightarrow p$

Please could someone give me some feed back on this proof? Does it look correct?

= $\lnot ((p\lor q)\land(p\lor(\lnot q)))\lor p$

= $(\lnot(p\lor q) \lor \lnot(p\lor(\lnot q)))\lor p$

= $((\lnot p\land \lnot q)\lor(\lnot p\land \lnot(\lnot q)))\lor p$

= $((\lnot p\land \lnot q)\lor(\lnot p\land q))\lor p$

= $(\lnot p(\lnot q \lor q)))\lor p$ <----- Is this step correct?

= $(\lnot p(T))\lor p$ $\equiv true$

Thanks a million

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@amWhy has already pointed out the actual error, but I’ll note in addition that the fourth line duplicates the third. –  Brian M. Scott Jan 17 '13 at 17:36
Unless you're required to stick with a particular proof formalism, a plain truth table proof might be shorter and simpler. –  Henning Makholm Jan 17 '13 at 18:07
(Though not shorter and simpler than amWhy's answer) –  Henning Makholm Jan 17 '13 at 18:15

The step in question should be, using the distributive law: $$((\lnot p\land \lnot q)\lor(\lnot p\land q))\lor p$$ $$\equiv (\lnot p \land (\lnot q \lor q)) \lor p)$$
$$\equiv (\lnot p \land T) \lor p$$ $$\equiv \lnot p \lor p \equiv T$$