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

$\blacksquare$