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Mirrors > Home > NFE Home > Th. List > df-sbc | Unicode version |
Description: Define the proper
substitution of a class for a set.
When
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 3048 below). For example, if
If we did not want to commit to any specific proper class behavior, we
could use this definition only to prove theorem dfsbcq 3048, which holds
for both our definition and Quine's, and from which we can derive a weaker
version of df-sbc 3047 in the form of sbc8g 3053. However, the behavior of
Quine's definition at proper classes is similarly arbitrary, and for
practical reasons (to avoid having to prove sethood of The theorem sbc2or 3054 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3048. The related definition df-csb 3137 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph |
. . 3
![]() ![]() | |
2 | vx |
. . 3
![]() ![]() | |
3 | cA |
. . 3
![]() ![]() | |
4 | 1, 2, 3 | wsbc 3046 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 1, 2 | cab 2339 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() |
6 | 3, 5 | wcel 1710 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 4, 6 | wb 176 |
1
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Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3048 dfsbcq2 3049 sbcex 3055 nfsbc1d 3063 nfsbcd 3066 cbvsbc 3074 sbcbid 3099 intab 3956 iotacl 4362 brab1 4684 |
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