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Definition df-sbc 2733
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 2757 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 2734 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 2734, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2733 in the form of sbc8g 2739. 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 2733 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 2732 . 2 wff [A / x]φ
51, 2cab 1999 . . 3 class {xφ}
63, 5wcel 1366 . 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  2734  dfsbcq2  2735  sbcex  2740  nfsbc1d  2748  nfsbcd  2751  cbvsbc  2759  sbcbid  2784  intab  3607  brab1  3772  iotacl  4805  riotasbc  5395  bdsbcALT  8244
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