Definition df-sbc 2951

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 2975 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 2952 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 2952, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2951 in the form of sbc8g 2957. 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 2951 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 2950	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2151	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2136	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 104	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2952  dfsbcq2  2953  sbcex  2958  nfsbc1d  2966  nfsbcd  2969  cbvsbcw  2977  cbvsbc  2978  sbcbi2  3000  sbcbid  3007  csbcow  3055  nfsbcdw  3078  intab  3852  brab1  4028  iotacl  5175  riotasbc  5812  bdsbcALT  13701