Modus Ponendo Ponens/Variant 3/Proof 1

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Theorem

$\vdash \paren {\paren {p \implies q} \land p} \implies q$


Proof

By the tableau method of natural deduction:

$\vdash \paren {\paren {p \implies q} \land p} \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \implies q} \land p$ Assumption (None)
2 1 $p \implies q$ Rule of Simplification: $\land \EE_1$ 1
3 1 $p$ Rule of Simplification: $\land \EE_2$ 1
4 1 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 3
5 $\paren {\paren {p \implies q} \land p} \implies q$ Rule of Implication: $\implies \II$ 1 – 4 Assumption 1 has been discharged

$\blacksquare$