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

*11. THEORY OF TWO APPARENT VARIABLES.

Summary of *11.
In this number, the propositions proved for one variable in *10 are to be
extended to two variables, with the addition of a few propositions having
no analogues for one variable, such as *11'2'21'23'24 and *11'53'55'6'7.
") (x, y)" stands for a proposition containing x and containing y; when
x and y are unassigned, 4 (x, y) is a propositional function of x and y. The
definition *1101 shows that "the truth of all values of (x, y)" does not
need to be taken as a new primitive idea, but is definable in terms of "the
truth of all values of rx." The reason is that, when x is assigned, ( (x, y)
becomes a function of one variable, namely y, whence it follows that, for
every possible value of x, "(y). ) (x, y) " embodies merely the primitive idea
introduced in *10. But "(y). (x, y)" is again only a function of one variable,
namely x, since y has here become an apparent variable. Hence the definition
*11'01 below is legitimate. We put:
*11'01. (x, y). ) (x, y). =:(x): (y). + (r, y) Df
*11-02. (x, y, z). 4 (x, y, z).=: (x):(y, z). (x, y, z) Df
*11-03. (gx, y). 4 (x, y). = (x): (My). 4 (x, y) Df
*11-04. (Ax, y, z). 4 (x, )y, z). =: (Ax): (My,,. (, y,z) Df
*11-05. (x, ). D,,,. J (x, y): =: (x, y): ) (x, y). ). f (x, y) Df
*11-06. < (x, y). -,y., (, y): =: (x, y): c (x, y). -. (x, y) Df
All the above definitions are supposed extended to any number of variables
that may occur.
The propositions of this section can all be extended to any finite number
of variables; as the analogy is exact, it is not necessary to carry the process
beyond two variables in our proofs.
In addition to the definition *1101, we need the primitive proposition
that "whatever possible argument x may be, 4 (x, y) is true whatever
possible argument y may be" implies the corresponding statement with x
and y interchanged. Either may be taken as the meaning of " 4 (x, y) is
true whatever possible arguments x and y may be."
The propositions of the present number are somewhat less used than
those of *10, but some of them are used frequently. Such are the following: