Definition df-sbc 2986

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 3011 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 2987 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 2987, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2986 in the form of sbc8g 2993. 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 2986 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 2985	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2179	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2164	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 105	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2987  dfsbcq2  2988  sbceqbid  2992  sbcex  2994  nfsbc1d  3002  nfsbcd  3005  cbvsbcw  3013  cbvsbc  3014  sbcbi2  3036  sbcbid  3043  csbcow  3091  nfsbcdw  3114  intab  3899  brab1  4076  iotacl  5239  riotasbc  5889  bdsbcALT  15351