Definition df-sbc 3682

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

When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3691). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3708 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 3683 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 3683, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3682 in the form of sbc8g 3689. 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 3682 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3690 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3683.

The related definition df-csb 3787 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 3681	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2758	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2050	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 198	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3683  dfsbcq2  3684  sbceqbid  3688  sbcex  3691  nfsbc1d  3699  nfsbcd  3702  cbvsbc  3710  sbcbid  3730  sbcbi2OLD  3735  intab  4779  brab1  4977  iotacl  6175  riotasbc  6952  scottexs  9110  scott0s  9111  hta  9120  issubc  16963  dmdprd  18870  sbceqbidf  30032  bnj1454  31767  bnj110  31783  setinds  32549  bj-csbsnlem  33718  rdgssun  34107  frege54cor1c  39630  frege55lem1c  39631  frege55c  39633