Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/153

*5·63.⁠$\scriptstyle {\vdash :.p\lor q.\equiv :p.\lor .\sim p.q}$$\scriptstyle {\left[*5\cdot 62{\frac {\sim p,q}{q,~~p}}\right]}$

*5·7. ⁠$\scriptstyle {\vdash :.p\lor r.\equiv .q\lor r:\equiv :r.\lor .p\equiv q}$$\scriptstyle {[*4\cdot 74.*1\cdot 3.*5\cdot 1.*4\cdot 37]}$

*5·71.⁠$\scriptstyle {\vdash :.q\supset \sim r.\supset :p\lor q.r.\equiv .p.r}$

In the following proof, as always henceforth, "$\scriptstyle {\text{Hp}}$" means the hypothesis of the proposition to be proved.

Dem.

${\begin{array}{lllr}\scriptstyle {\vdash .*4\cdot 4.}&\scriptstyle {\supset \vdash :.p\lor q}&\scriptstyle {.r.\equiv :p.r.\lor .q.r}&\scriptstyle {\qquad \qquad {\text{(1)}}}\\\scriptstyle {\vdash .*4\cdot 62\cdot 51.}&\scriptstyle {\supset \vdash ::{\text{Hp}}.}&\scriptstyle {\supset :.\sim (q.r):.}\\\scriptstyle {[*4\cdot 74]}&&\scriptstyle {\supset :.p.r.\lor .q.r:\equiv :p.r}&\scriptstyle {\text{(2)}}\\\scriptstyle {\vdash .(1).(2).}&\scriptstyle {*4\cdot 22.\supset \vdash }&\scriptstyle {.{\text{Prop}}}\end{array}}$

*5·74.⁠$\scriptstyle {\vdash :.p.\supset .q\equiv r:\equiv :p\supset q.\equiv .p\supset r}$

Dem.

${\begin{array}{lllr}\scriptstyle {\vdash .*5\cdot 41.\supset \vdash ::p\supset q.\supset .}&\scriptstyle {p\supset r:\equiv :p}&\scriptstyle {.\supset .q\supset r:.}\\&\scriptstyle {p\supset r.\supset .p}&\scriptstyle {\supset q:\equiv :p.\supset .r\supset q}&\scriptstyle {\qquad {\text{(1)}}}\\\scriptstyle {\vdash .(1).*4\cdot 38.\supset \vdash ::p\supset q.}&\scriptstyle {\equiv .p\supset r.}&\scriptstyle {\equiv :.p.\supset .q\supset r:p.\supset .r\supset q:.}\\\scriptstyle {[*4\cdot 76]}&&\scriptstyle {\equiv :.p.\supset .q\equiv r::\supset \vdash .{\text{Prop}}}\end{array}}$

*5·75.⁠$\scriptstyle {\vdash :.r\supset \sim q:p.\equiv .q\lor r:\supset :p.\sim q.\equiv .r}$

Dem.

${\begin{array}{llllr}\scriptstyle {\vdash .*5\cdot 6.}&\scriptstyle {\supset \vdash :.{\text{Hp}}.}&\scriptstyle {\supset :p.\sim q}&\scriptstyle {.\supset .r}&\scriptstyle {\qquad \qquad {\text{(1)}}}\\\scriptstyle {\vdash .*3\cdot 27.}&\scriptstyle {\supset \vdash :.{\text{Hp}}.}&\scriptstyle {\supset :q\lor r.}&\scriptstyle {\supset .p:}\\\scriptstyle {[*4\cdot 77]}&&\scriptstyle {\supset :r\supset p}&&\scriptstyle {\text{(2)}}\\\scriptstyle {\vdash .*3\cdot 26.}&&\scriptstyle {\supset \vdash :.{\text{Hp}}.}&\scriptstyle {\supset :r\supset \sim q}&\scriptstyle {\text{(3)}}\\\scriptstyle {\vdash .(2).(3)}&\scriptstyle {.{\text{Comp}}.}&\scriptstyle {\supset \vdash :.{\text{Hp}}.}&\scriptstyle {\supset :r\supset p.r\supset \sim q:}\\\scriptstyle {[{\text{Comp}}]}&&\scriptstyle {\supset :r.\supset .p.\sim q}&&\scriptstyle {\text{(4)}}\\\scriptstyle {\vdash .(1).(4)}&\scriptstyle {.{\text{Comp}}.}&\scriptstyle {\supset \vdash :.{\text{Hp}}.}&\scriptstyle {\supset :p.\sim q.\equiv .r:.\supset \vdash .{\text{Prop}}}\end{array}}$