Definition df-sbc 2965

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 2990 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 2966 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 2966, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2965 in the form of sbc8g 2972. 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 2965 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 2964	. 2 wff [𝐴 / 𝑥]𝜑
5	1, 2	cab 2163	. . 3 class {𝑥 ∣ 𝜑}
6	3, 5	wcel 2148	. 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑}
7	4, 6	wb 105	1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})

This definition is referenced by:  dfsbcq  2966  dfsbcq2  2967  sbceqbid  2971  sbcex  2973  nfsbc1d  2981  nfsbcd  2984  cbvsbcw  2992  cbvsbc  2993  sbcbi2  3015  sbcbid  3022  csbcow  3070  nfsbcdw  3093  intab  3875  brab1  4052  iotacl  5203  riotasbc  5848  bdsbcALT  14696