Definition df-sbc 3777

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3786). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3808 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 3778 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 3778, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3777 in the form of sbc8g 3784. 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 3777 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

Theorem sbc2or 3785 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3778.

The related definition df-csb 3893 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 3776	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2710	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2107	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 205	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3778  dfsbcq2  3779  sbceqbid  3783  sbcex  3786  nfsbc1d  3794  nfsbcdw  3797  nfsbcd  3800  sbc5  3804  sbc6g  3806  cbvsbcw  3810  cbvsbcvw  3811  cbvsbc  3812  sbcieg  3816  sbcied  3821  sbcbid  3834  sbcbi2  3838  sbcimdv  3850  sbcg  3855  intab  4981  brab1  5195  iotacl  6526  riotasbc  7379  scottexs  9878  scott0s  9879  hta  9888  issubc  17781  dmdprd  19860  sbceqbidf  31705  bnj1454  33791  bnj110  33807  setinds  34688  bj-csbsnlem  35721  rdgssun  36197  frege54cor1c  42599  frege55lem1c  42600  frege55c  42602