Definition df-sb 1881

Description: Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [ y / x ] ph to mean "the wff that results from the proper substitution of y for x in the wff ph." That is, y properly replaces x. For example, [ x / y ] z e. y is the same as z e. x, as shown in elsb4 2435. We can also use [ y / x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2353.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when y is properly substituted for x in ph ( x )." For example, if the original ph ( x ) is x = y, then ph ( y ) is y = y, from which we obtain that ph ( x ) is x = x. So what exactly does ph ( x ) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2376, sbcom2 2445 and sbid2v 2457).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 2114 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2455 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2374. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 2430 and sb6 2429.

There are no restrictions on any of the variables, including what variables may occur in wff ph. (Contributed by NM, 10-May-1993.)

Assertion

Ref	Expression
df-sb	|- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )

Detailed syntax breakdown of Definition df-sb

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4	1, 2, 3	wsb 1880	. 2 wff [ y / x ] ph
5	2, 3	weq 1874	. . . 4 wff x = y
6	5, 1	wi 4	. . 3 wff ( x = y -> ph )
7	5, 1	wa 384	. . . 4 wff ( x = y /\ ph )
8	7, 2	wex 1704	. . 3 wff E. x ( x = y /\ ph )
9	6, 8	wa 384	. 2 wff ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) )
10	4, 9	wb 196	1 wff ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )

This definition is referenced by:  sbequ2  1882  sb1  1883  sbequ8  1885  sbimi  1886  sbequ1  2110  sb2  2352  drsb1  2377  sbn  2391  subsym1  32426  bj-sb2v  32753  bj-dfsb2  32825  frege55b  38191