Rule of Simplification/Sequent Form/Formulation 2/Proof 1/Form 1
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Theorem
- $\vdash p \land q \implies p$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \land q$ | Assumption | (None) | ||
2 | 1 | $p$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
3 | $p \land q \implies p$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $41$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercises $1.4: \ 1 \ \text{(a)}$