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Definition df-sbc 2759
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 2783 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 2760 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 2760, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2759 in the form of sbc8g 2765. 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 2759 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.)

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 2758 . 2 wff [A / x]φ
51, 2cab 2023 . . 3 class {xφ}
63, 5wcel 1390 . 2 wff A {xφ}
74, 6wb 98 1 wff ([A / x]φA {xφ})
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  2760  dfsbcq2  2761  sbcex  2766  nfsbc1d  2774  nfsbcd  2777  cbvsbc  2785  sbcbid  2810  intab  3635  brab1  3800  iotacl  4833  riotasbc  5426  bdsbcALT  9248
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