Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION A]

EQUIVALENCE AND FORMAL RULES

127

*4'73. F:.q.:)p.=-p.q [Simp.*4'71]
This proposition is very useful, since it shows that a true factor may be
omitted from a product without altering its truth or falsehood, just as a true
hypothesis may be omitted from an implication.
*4-74. F:. p. D: (.. pvq [*2-21. 4-72]
*4'76D q:..p -41 --:p. (*1-01)
*4'77. F:.qDp.rDp.-:qvr. D.p [*3-44. Add. *22]
*4'78. F:.pDq.v.por:-:p. ).qvr
Dem.
F.,4'2. (1'01). 1F:.p D. v. p ) r: -: p- q. v. ~pv r:
[*4'33] =. ~p. v. q v,p v r:
[*4'31'37] -: ~p.v.,pvqvr:
[*4'33] -: ~pv~p. v. vr:
[*4'25'37] =: rp.. q v r:
[~4'2.(,1'01)] =: p,. D v:. q v r. D. Prop
*4'79. H.qp. v.r p.q.r. r ).
Demn.
F. 4'1'39.D:.q) p.v.rDp:-=: ~. v. pD r:
[*4-78]:.. r:
[*2'15]: ~(~ qv~. ).:
[*4.2.(*3.01)] -q. r. p:.. p.. Prop
Note. The analogues, for classes, of *4'78'79 are false. Take, e.g. *4'78,
and put p = English people, q = men, r = women. Then p is contained in q
or r, but is not contained in q and is not contained in r.
*4'8. F:p ) p.-. ~ [*2'01.Simp]
*4'81. k: ~ p Dp. -. p [*2-18. Simp]
*4-82. F:p q. )c ~ q. -. p [*2'65.Imp. *221. Comp]
*4-83. F: p q. g p q.= - [*2'61. Imp. Simp. Comp]
Note. *4'82'83 may also be obtained from *4'43, of which they are
virtually other forms.
*4'84. F:.p-q. D:pDr.. q )r [*2-06..3-47]
*4-85. F:.p=-q.:r)p.=.r)q [*2-05. *3'47]
*4-86. F:.p=q.D:p=r.-.q-r [*4'21-22]
*4'87. k:. p. q.. r:- p. q r:: q.. p ): q. p.. r
[Exp. Comm. Imp]
*4'87 embodies in one proposition the principles of exportation and
importation and the commutative principle.