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Definition df-sbc 2738
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 2762 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 2739 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 2739, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2738 in the form of sbc8g 2744. 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 2738 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 2737 . 2 wff [A / x]φ
51, 2cab 2004 . . 3 class {xφ}
63, 5wcel 1370 . 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  2739  dfsbcq2  2740  sbcex  2745  nfsbc1d  2753  nfsbcd  2756  cbvsbc  2764  sbcbid  2789  intab  3614  brab1  3779  iotacl  4813  riotasbc  5403  bdsbcALT  7225
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