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Definition df-sbc 2817
 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 2841 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 2818 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 2818, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2817 in the form of sbc8g 2823. 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 2817 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 2816 . 2 wff [𝐴 / 𝑥]𝜑
51, 2cab 2068 . . 3 class {𝑥𝜑}
63, 5wcel 1434 . 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  2818  dfsbcq2  2819  sbcex  2824  nfsbc1d  2832  nfsbcd  2835  cbvsbc  2843  sbcbi2  2865  sbcbid  2872  intab  3673  brab1  3838  iotacl  4920  riotasbc  5514  bdsbcALT  10808
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