Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/134

112 $$\scriptstyle{\vdash:\sim(p\or q).\supset.\sim p\or q\qquad\left[*2\cdot45.*2\cdot2\frac{\sim p}{p}.\text{Syll}\right]}$$

$$\scriptstyle{\vdash:\sim(p\or q).\supset.p\or\sim q\qquad\left[*2\cdot46.*1\cdot3\frac{\sim q}{q}.\text{Syll}\right]}$$

$$\scriptstyle{\vdash:\sim(p\or q).\supset.\sim p\or\sim q\qquad\left[*2\cdot45.*2\cdot2\frac{\sim p,\sim q}{p,q}.\text{Syll}\right]}$$

$$\scriptstyle{\vdash:\sim(p\supset q).\supset.\sim p\supset q\qquad\left[*2\cdot47\frac{\sim p}{p}\right]}$$

$$\scriptstyle{\vdash:\sim(p\supset q).\supset.p\supset\sim q\qquad\left[*2\cdot48\frac{\sim p}{p}\right]}$$

$$\scriptstyle{\vdash:\sim(p\supset q).\supset.p\supset\sim q\qquad\left[*2\cdot49\frac{\sim p}{p}\right]}$$

$$\scriptstyle{\vdash:\sim(p\supset q).\supset.q\supset p\qquad[*2\cdot52\cdot17]}$$

$$\scriptstyle{\vdash:p\or q.\supset.\sim p\supset q}$$

Dem.

$$\scriptstyle{\vdash:\sim p\supset q.\supset.p\or q\qquad[*2\cdot14\cdot38]}$$

$$\scriptstyle{\vdash:.\sim p.\supset:p\or q.\supset.q\quad[*2\cdot53.\text{Comm}]}$$

$$\scriptstyle{\vdash:.\sim q.\supset:p\or q.\supset.p\quad\left[*2\cdot55\frac{q,p}{p,q}.\text{Perm}\right]}$$

$$\scriptstyle{\vdash:.\sim p\supset q.\supset:p\supset q.\supset.q}$$

Dem.

$$\scriptstyle{\vdash:.p\supset q.\supset:\sim p\supset q.\supset.q\quad[*2\cdot6.\text{Comm}]}$$

$$\scriptstyle{\vdash:.p\supset q.\supset:\sim p\supset q.\supset.q\qquad[*2\cdot6.\text{Comm}]}$$

$$\scriptstyle{\vdash:.p\or q.\supset:p\supset q.\supset.q\qquad[*2\cdot53\cdot6.\text{Syll}]}$$

$$\scriptstyle{\vdash:.p\supset q.\supset:p\or q.\supset.q\qquad[*2\cdot62.\text{Comm}]}$$

$$\scriptstyle{\vdash:.p\or q.\supset:\sim p\or q.\supset.q\qquad[*2\cdot62]}$$

$$\scriptstyle{\vdash:.p\or q.\supset:p\or\sim q.\supset.p\qquad\left[*2\cdot63\frac{q,p}{p,q}.\text{Perm}\right]}$$

$$\scriptstyle{\vdash:.p\supset q.\supset:p\supset\sim q.\supset.\sim p\quad\left[*2\cdot64\frac{\sim p}{p}\right]}$$

$$\scriptstyle{\vdash:.p\or q.\supset.q:\supset.p\supset q}$$

Dem.