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

Definition df-sbc 2765
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 2789 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 2766 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of 𝜑, 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 2766, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2765 in the form of sbc8g 2771. 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 2765 and assert that [𝐴 / 𝑥]𝜑 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]𝜑𝐴 ∈ {𝑥𝜑})

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 cA . . 3 class 𝐴
41, 2, 3wsbc 2764 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2026 . . 3 class {𝑥𝜑}
63, 5wcel 1393 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 98 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2766  dfsbcq2  2767  sbcex  2772  nfsbc1d  2780  nfsbcd  2783  cbvsbc  2791  sbcbid  2816  intab  3644  brab1  3809  iotacl  4890  riotasbc  5483  bdsbcALT  9979
  Copyright terms: Public domain W3C validator