Definition df-sbc 3052

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 3077 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 3053 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 3053, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3052 in the form of sbc8g 3059. 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 3052 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 3051	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2224	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2209	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 105	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  3053  dfsbcq2  3054  sbceqbid  3058  sbcex  3060  nfsbc1d  3068  nfsbcd  3071  cbvsbcw  3079  cbvsbc  3080  sbcbi2  3102  sbcbid  3109  csbcow  3158  nfsbcdw  3181  intab  3997  brab1  4176  iotacl  5360  riotasbc  6048  bdsbcALT  16802