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Definition df-sb 1660
 Description: Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2092. 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, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does 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 2113, sbcom2 2192 and sbid2v 2202). Note that our definition is valid even when and are replaced with the same variable, as sbid 1948 shows. We achieve this by having 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 2200 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 2111. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2178 and sb6 2177. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
df-sb

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 vy . . 3
41, 2, 3wsb 1659 . 2
52, 3weq 1654 . . . 4
65, 1wi 4 . . 3
75, 1wa 360 . . . 4
87, 2wex 1551 . . 3
96, 8wa 360 . 2
104, 9wb 178 1
 Colors of variables: wff set class This definition is referenced by:  sbequ2  1661  sbequ2OLD  1662  sb1  1663  sbimi  1665  sbequ1  1944  sb2  2091  drsb1  2114  sbn  2132  sbnOLD  2133  sb6  2177  subsym1  26179  drsb1NEW7  29568  sb2NEW7  29599  sbnNEW7  29624  sb6NEW7  29659
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