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Theorem dfsbcq 3048
 Description: This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3047 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3049 instead of df-sbc 3047. (dfsbcq2 3049 is needed because unlike Quine we do not overload the df-sb 1649 syntax.) As a consequence of these theorems, we can derive sbc8g 3053, which is a weaker version of df-sbc 3047 that leaves substitution undefined when is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3053, so we will allow direct use of df-sbc 3047 after theorem sbc2or 3054 below. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)
Assertion
Ref Expression
dfsbcq

Proof of Theorem dfsbcq
StepHypRef Expression
1 eleq1 2413 . 2
2 df-sbc 3047 . 2
3 df-sbc 3047 . 2
41, 2, 33bitr4g 279 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wceq 1642   wcel 1710  cab 2339  wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-sbc 3047 This theorem is referenced by:  sbceq1d  3051  sbc8g  3053  spsbc  3058  sbcco  3068  sbcco2  3069  sbcie2g  3079  elrabsf  3084  eqsbc3  3085  csbeq1  3139  sbcnestgf  3183  sbcco3g  3191  cbvralcsf  3198
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