Definition df-sbc 2956

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 2980 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 2957 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 2957, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2956 in the form of sbc8g 2962. 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 2956 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 2955	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2156	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2141	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 104	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2957  dfsbcq2  2958  sbcex  2963  nfsbc1d  2971  nfsbcd  2974  cbvsbcw  2982  cbvsbc  2983  sbcbi2  3005  sbcbid  3012  csbcow  3060  nfsbcdw  3083  intab  3860  brab1  4036  iotacl  5183  riotasbc  5824  bdsbcALT  13894