Principle of Composition/Formulation 1/Forward Implication

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Theorem

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


Proof

By the tableau method of natural deduction:

$\paren {p \implies r} \lor \paren {q \implies r} \vdash \paren {p \land q} \implies r$
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \implies r} \lor \paren {q \implies r}$ Premise (None)
2 2 $p \implies r$ Assumption (None)
3 3 $p \land q$ Assumption (None)
4 3 $p$ Rule of Simplification: $\land \mathcal E_ 1$ 3
5 2, 3 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 2 , 4
6 2 $\paren {p \land q} \implies r$ Rule of Implication: $\implies \mathcal I$ 3 – 5 Assumption 3 has been discharged
7 7 $q \implies r$ Assumption (None)
8 8 $p \land q$ Assumption (None)
9 8 $q$ Rule of Simplification: $\land \mathcal E_ 2$ 8
10 7, 8 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 7 , 9
11 7 $\paren {p \land q} \implies r$ Rule of Implication: $\implies \mathcal I$ 8 – 10 Assumption 8 has been discharged
12 1 $\paren {p \land q} \implies r$ Proof by Cases: $\text{PBC}$ 1 , 2 – 6 , 7 – 11 Assumptions 2  and 7 have been discharged

$\blacksquare$