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Definition df-sbc 2830
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 2854 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 2831 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 2831, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2830 in the form of sbc8g 2836. 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 2830 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]𝜑𝐴 ∈ {𝑥𝜑})

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 cA . . 3 class 𝐴
41, 2, 3wsbc 2829 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2071 . . 3 class {𝑥𝜑}
63, 5wcel 1436 . 2 wff 𝐴 ∈ {𝑥𝜑}
74, 6wb 103 1 wff ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2831  dfsbcq2  2832  sbcex  2837  nfsbc1d  2845  nfsbcd  2848  cbvsbc  2856  sbcbi2  2878  sbcbid  2885  intab  3702  brab1  3867  iotacl  4971  riotasbc  5586  bdsbcALT  11219
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