Principle of Composition/Formulation 1/Reverse Implication

Theorem

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


Proof

By the tableau method of natural deduction:

$\paren {p \land q} \implies r \vdash \paren {p \implies r} \lor \paren {q \implies r} $
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \land q} \implies r$ Premise (None)
2 1 $\neg \paren {p \lor q} \lor r$ Sequent Introduction 1 Rule of Material Implication
3 1 $\neg p \lor \neg q \lor r$ Sequent Introduction 2 De Morgan's Laws: Disjunction of Negations
4 1 $r \lor \neg p \lor \neg q \lor r$ Rule of Addition: $\lor \II_ 2$ 3
5 1 $\neg p \lor r \lor \neg q \lor r$ Sequent Introduction 4 Disjunction is Commutative
6 1 $\paren {p \implies r} \lor \neg q \lor r$ Sequent Introduction 5 Rule of Material Implication
7 1 $\paren {p \implies r} \lor \paren {q \implies r}$ Sequent Introduction 6 Rule of Material Implication

$\blacksquare$

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