Definition df-sb 1686

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 1698.

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 1761, sbcom2 1904 and sbid2v 1913).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1697 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 1908 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 1911.

When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1808 and sb6 1807.

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 1685	. 2 wff [ y / x ] ph
5	2, 3	weq 1432	. . . 4 wff x = y
6	5, 1	wi 4	. . 3 wff ( x = y -> ph )
7	5, 1	wa 102	. . . 4 wff ( x = y /\ ph )
8	7, 2	wex 1421	. . 3 wff E. x ( x = y /\ ph )
9	6, 8	wa 102	. 2 wff ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) )
10	4, 9	wb 103	1 wff ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )

This definition is referenced by:  sbimi  1687  sb1  1689  sb2  1690  sbequ1  1691  sbequ2  1692  drsb1  1720  spsbim  1764  sbequ8  1768  sbidm  1772  sb6  1807  hbsbv  1858