Rule of Simplification/Sequent Form/Formulation 2/Proof 1

Theorem

$(1): \quad \vdash p \land q \implies p$

$(2): \quad \vdash p \land q \implies q$

Proof

Form 1

By the tableau method of natural deduction:

$\vdash p \land q \implies p$

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

$\blacksquare$

Form 2

By the tableau method of natural deduction:

$\vdash p \land q \implies q$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \land q$	Assumption	(None)
2		1	$q$	Rule of Simplification: $\land \EE_2$	1
3			$p \land q \implies q$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$