Definition df-sbc 2863

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 2887 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 2864 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 2864, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2863 in the form of sbc8g 2869. 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 2863 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 2862	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2086	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 1448	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 104	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2864  dfsbcq2  2865  sbcex  2870  nfsbc1d  2878  nfsbcd  2881  cbvsbc  2889  sbcbi2  2911  sbcbid  2918  intab  3747  brab1  3920  iotacl  5047  riotasbc  5677  bdsbcALT  12638