Proof By Direct Proof

Assume $P$ is true, then conclude that $Q$ is true.

Format

If you have the form: $(P_1\wedge P_2\wedge \dots P_n)⟹Q$

1. Suppose all premises P are true
2. … Use Logical Equivalence Laws and Rules of Inference to prove that $Q$
3. Q

Examples

$\forall x,y((x>3)\wedge (y<2))⟹(x^2-2y>5)$

1. Take arbitrary $x,y$
2. Suppose x > 3
3. Suppose y < 2
4. $x^2=x\ast x>3\ast x>3\ast 3=9$ 2. by mult. by pos
5. $-2y>-4$ 3. multiply by negative
6. $x^2-2y>9-4=5$ 4,5 add
7. $((x>3)\wedge (y<2)⟹(x^2-2y>5))$ 2,3,6 implication
8. $\forall x\forall y,(x>3)\wedge (y<2)⟹x^2-2y>5$ 1,7 universal generalization