Definition df-sb 1155

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 1168.

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 remarkable little formula that is exactly equivalent and gives us a single direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1213, sbcom2 1316 and sbid2v 1325).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1167 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 sb7 1322 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 1210. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1252 and sb6 1251.

There are no restrictions on any of the variables, including what variables may occur in wff ph.

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 set x
3		vy	. . . 4 set y
4	3	cv 1098	. . 3 class y
5	1, 2, 4	wsbc 1153	. 2 wff [y / x]ph
6	2	cv 1098	. . . . 5 class x
7	6, 4	wceq 1099	. . . 4 wff x = y
8	7, 1	wi 3	. . 3 wff (x = y -> ph)
9	7, 1	wa 223	. . . 4 wff (x = y /\ ph)
10	9, 2	wex 956	. . 3 wff E.x(x = y /\ ph)
11	8, 10	wa 223	. 2 wff ((x = y -> ph) /\ E.x(x = y /\ ph))
12	5, 11	wb 146	1 wff ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))

This definition is referenced by:  sbimi 1156  drsb1 1158  sb1 1159  sb2 1160  sbequ1 1161  sbequ2 1162  sbn 1215  sb6 1251

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