ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-sbc Unicode version

Definition df-sbc 2787
Description: Define the proper substitution of a class for a set.

When  A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2811 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2788 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of  ph, and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 2788, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2787 in the form of sbc8g 2793. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of  A in every use of this definition) we allow direct reference to df-sbc 2787 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Assertion
Ref Expression
df-sbc  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  setvar  x
3 cA . . 3  class  A
41, 2, 3wsbc 2786 . 2  wff  [. A  /  x ]. ph
51, 2cab 2042 . . 3  class  { x  |  ph }
63, 5wcel 1409 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 102 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2788  dfsbcq2  2789  sbcex  2794  nfsbc1d  2802  nfsbcd  2805  cbvsbc  2813  sbcbi2  2835  sbcbid  2842  intab  3671  brab1  3836  iotacl  4917  riotasbc  5510  bdsbcALT  10345
  Copyright terms: Public domain W3C validator