Definition df-sbc 3423

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 3449 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 3424 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 𝐴 rather than our falsehood. (This can be seen by substituting 𝐴, 𝑦, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic 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 3424, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3423 in the form of sbc8g 3430. 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 3423 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3431 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3424.

The related definition df-csb 3520 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]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

Detailed syntax breakdown of Definition df-sbc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		cA	. . 3 class 𝐴
4	1, 2, 3	wsbc 3422	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2607	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 1987	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 196	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3424  dfsbcq2  3425  sbceqbid  3429  sbcex  3432  nfsbc1d  3440  nfsbcd  3443  cbvsbc  3451  sbcbi2  3471  sbcbid  3476  intab  4479  brab1  4670  iotacl  5843  riotasbc  6591  scottexs  8710  scott0s  8711  hta  8720  issubc  16435  dmdprd  18337  sbceqbidf  29210  bnj1454  30673  bnj110  30689  setinds  31437  bj-csbsnlem  32598  frege54cor1c  37730  frege55lem1c  37731  frege55c  37733