Definition df-sbc 2914

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 2938 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 2915 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 2915, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2914 in the form of sbc8g 2920. 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 2914 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 2913	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2126	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 1481	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 104	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2915  dfsbcq2  2916  sbcex  2921  nfsbc1d  2929  nfsbcd  2932  cbvsbcw  2940  cbvsbc  2941  sbcbi2  2963  sbcbid  2970  intab  3808  brab1  3983  iotacl  5119  riotasbc  5753  bdsbcALT  13228