Rule of Exportation/Reverse Implication/Formulation 1/Proof
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Theorem
- $p \implies \paren {q \implies r} \vdash \paren {p \land q} \implies r$
Proof
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies \paren {q \implies r}$ | Premise | (None) | ||
2 | 2 | $p \land q$ | Assumption | (None) | ||
3 | 2 | $p$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
4 | 2 | $q$ | Rule of Simplification: $\land \EE_2$ | 2 | ||
5 | 1, 2 | $q \implies r$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 1, 3 | ||
6 | 1, 2 | $r$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 5, 4 | ||
7 | 1 | $\paren {p \land q} \implies r$ | Rule of Implication: $\implies \II$ | 2 – 6 | Assumption 2 has been discharged |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $3$ Conjunction and Disjunction: Theorem $16$
- 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: Example $1.14$