Definition df-sbc 3046

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 3071 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 3047 below). Unfortunately, Quine's definition requires a recursive syntactical 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 3047, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3046 in the form of sbc8g 3053. 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 3046 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The related definition df-csb 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 3045	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2220	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2205	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 105	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3047  dfsbcq2  3048  sbceqbid  3052  sbcex  3054  nfsbc1d  3062  nfsbcd  3065  cbvsbcw  3073  cbvsbc  3074  sbcbi2  3096  sbcbid  3103  csbcow  3152  nfsbcdw  3175  intab  3983  brab1  4162  iotacl  5342  riotasbc  6028  bdsbcALT  16755