Factor Principles/Conjunction on Left/Formulation 2
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Theorem
- $\vdash \paren {p \implies q} \implies \paren {\paren {r \land p} \implies \paren {r \land q} }$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies q$ | Assumption | (None) | ||
2 | 1 | $\paren {\paren {r \land p} \implies \paren {r \land q} }$ | Sequent Introduction | 1 | Factor Principles: Conjunction on Left: Formulation 1 | |
3 | 1 | $\paren {p \implies q} \implies \paren {\paren {r \land p} \implies \paren {r \land q} }$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T30}$