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Definition df-sb 1630
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 1964.

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 2000, sbcom2 2053 and sbid2v 2062).

Note that our definition is valid even when  x and  y are replaced with the same variable, as sbid 1863 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 2058 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 1996. When  x and  y are distinct, we can express proper substitution with the simpler expressions of sb5 2039 and sb6 2038.

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 1629 . 2  wff  [ y  /  x ] ph
52, 3weq 1624 . . . 4  wff  x  =  y
65, 1wi 4 . . 3  wff  ( x  =  y  ->  ph )
75, 1wa 358 . . . 4  wff  ( x  =  y  /\  ph )
87, 2wex 1528 . . 3  wff  E. x
( x  =  y  /\  ph )
96, 8wa 358 . 2  wff  ( ( x  =  y  ->  ph )  /\  E. x
( x  =  y  /\  ph ) )
104, 9wb 176 1  wff  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
Colors of variables: wff set class
This definition is referenced by:  sbequ2  1631  sb1  1632  sbimi  1633  sbequ1  1859  drsb1  1962  sb2  1963  sbn  2002  sb6  2038  subsym1  24866
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