Definition df-sbc 3569

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 3595 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 3570 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 3570, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3569 in the form of sbc8g 3576. 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 3569 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3577 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3570.

The related definition df-csb 3667 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 3568	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2738	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2131	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 196	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3570  dfsbcq2  3571  sbceqbid  3575  sbcex  3578  nfsbc1d  3586  nfsbcd  3589  cbvsbc  3597  sbcbi2  3617  sbcbid  3622  intab  4651  brab1  4844  iotacl  6027  riotasbc  6781  scottexs  8915  scott0s  8916  hta  8925  issubc  16688  dmdprd  18589  sbceqbidf  29622  bnj1454  31211  bnj110  31227  setinds  31980  bj-csbsnlem  33196  frege54cor1c  38703  frege55lem1c  38704  frege55c  38706