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Definition df-sb 1884
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 from the proper substitution of  y 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 1897.

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 1953, sbcom2 2077 and sbid2v 2086).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 1896 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 2082 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 1949. When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 1994 and sb6 1993.

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
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
3 vy . . 3  set  y
41, 2, 3wsb 1883 . 2  wff  [ y  /  x ] ph
52, 3weq 1620 . . . 4  wff  x  =  y
65, 1wi 6 . . 3  wff  ( x  =  y  ->  ph )
75, 1wa 360 . . . 4  wff  ( x  =  y  /\  ph )
87, 2wex 1537 . . 3  wff  E. x
( x  =  y  /\  ph )
96, 8wa 360 . 2  wff  ( ( x  =  y  ->  ph )  /\  E. x
( x  =  y  /\  ph ) )
104, 9wb 178 1  wff  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
Colors of variables: wff set class
This definition is referenced by:  sbimi  1885  drsb1  1887  sb1  1888  sb2  1889  sbequ1  1890  sbequ2  1891  sbn  1955  sb6  1993  subsym1  24227
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