Definition df-sbc 3721

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3730). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3750 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 3722 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 3722, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3721 in the form of sbc8g 3728. 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 3721 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3729 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3722.

The related definition df-csb 3829 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 3720	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2776	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2111	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 209	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3722  dfsbcq2  3723  sbceqbid  3727  sbcex  3730  nfsbc1d  3738  nfsbcdw  3741  nfsbcd  3744  cbvsbcw  3752  cbvsbc  3754  sbcbid  3773  sbcbi2  3778  intab  4868  brab1  5078  iotacl  6310  riotasbc  7111  scottexs  9300  scott0s  9301  hta  9310  issubc  17097  dmdprd  19113  sbceqbidf  30257  bnj1454  32224  bnj110  32240  setinds  33136  bj-csbsnlem  34344  rdgssun  34795  frege54cor1c  40616  frege55lem1c  40617  frege55c  40619