Definition df-sb 1743

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 when y is properly substituted for x in the wff ph." We can also use [ y / x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1755.

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 1820, sbcom2 1967 and sbid2v 1976).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1754 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 1971 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1974.

When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1867 and sb6 1866.

In classical logic, another possible definition is ( x = y /\ ph ) \/ A. x ( x = y -> ph ) but we do not have an intuitionistic proof that this is equivalent.

There are no restrictions on any of the variables, including what variables may occur in wff ph. (Contributed by NM, 5-Aug-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 1742	. 2 wff [ y / x ] ph
5	2, 3	weq 1483	. . . 4 wff x = y
6	5, 1	wi 4	. . 3 wff ( x = y -> ph )
7	5, 1	wa 103	. . . 4 wff ( x = y /\ ph )
8	7, 2	wex 1472	. . 3 wff E. x ( x = y /\ ph )
9	6, 8	wa 103	. 2 wff ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) )
10	4, 9	wb 104	1 wff ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )

This definition is referenced by:  sbimi  1744  sb1  1746  sb2  1747  sbequ1  1748  sbequ2  1749  drsb1  1779  spsbim  1823  sbequ8  1827  sbidm  1831  sb6  1866  hbsbv  1921  nfsbv  1927