Definition df-sbc 3773

Description: Define the proper substitution of a class for a set.

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3782). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3802 for our definition, whose right-hand side 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 3774 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 3774, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3773 in the form of sbc8g 3780. 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 3773 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3781 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3774.

The related definition df-csb 3884 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 3772	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2799	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2114	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 208	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3774  dfsbcq2  3775  sbceqbid  3779  sbcex  3782  nfsbc1d  3790  nfsbcdw  3793  nfsbcd  3796  cbvsbcw  3804  cbvsbc  3806  sbcbid  3826  sbcbi2OLD  3832  intab  4906  brab1  5114  iotacl  6341  riotasbc  7132  scottexs  9316  scott0s  9317  hta  9326  issubc  17105  dmdprd  19120  sbceqbidf  30250  bnj1454  32114  bnj110  32130  setinds  33023  bj-csbsnlem  34223  rdgssun  34662  frege54cor1c  40310  frege55lem1c  40311  frege55c  40313