Definition df-sb 1619

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 1631.

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 1694, sbcom2 1836 and sbid2v 1845).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1630 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 1840 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 1843.

When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1740 and sb6 1739.

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 1618	. 2 wff [y / x]ph
5	2, 3	weq 1365	. . . 4 wff x = y
6	5, 1	wi 4	. . 3 wff (x = y -> ph)
7	5, 1	wa 97	. . . 4 wff (x = y /\ ph)
8	7, 2	wex 1354	. . 3 wff E.x(x = y /\ ph)
9	6, 8	wa 97	. 2 wff ((x = y -> ph) /\ E.x(x = y /\ ph))
10	4, 9	wb 98	1 wff ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))

This definition is referenced by:  sbimi  1620  sb1  1622  sb2  1623  sbequ1  1624  sbequ2  1625  drsb1  1653  spsbim  1697  sbequ8  1700  sbidm  1704  sb6  1739  hbsbv  1790