Factor Principles/Conjunction on Left/Formulation 1/Proof 1
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Theorem
- $p \implies q \vdash \paren {r \land p} \implies \paren {r \land q}$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies q$ | Premise | (None) | ||
2 | $r \implies r$ | Law of Identity | (None) | This is a theorem so depends on nothing | ||
3 | 1 | $\paren {r \implies r} \land \paren {p \implies q}$ | Rule of Conjunction: $\land \II$ | 2, 1 | ||
4 | 1 | $\paren {r \land p} \implies \paren {r \land q}$ | Sequent Introduction | 3 | Praeclarum Theorema |
$\blacksquare$