Definition df-sbc 3001

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 3026 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 3002 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 3002, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3001 in the form of sbc8g 3008. 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 3001 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 3000	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2192	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2177	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 105	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3002  dfsbcq2  3003  sbceqbid  3007  sbcex  3009  nfsbc1d  3017  nfsbcd  3020  cbvsbcw  3028  cbvsbc  3029  sbcbi2  3051  sbcbid  3058  csbcow  3106  nfsbcdw  3129  intab  3917  brab1  4096  iotacl  5262  riotasbc  5925  bdsbcALT  15909