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Theorem dfsbcq 2743
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 2742 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 2744 instead of df-sbc 2742. (dfsbcq2 2744 is needed because unlike Quine we do not overload the df-sb 1628 syntax.) As a consequence of these theorems, we can derive sbc8g 2748, which is a weaker version of df-sbc 2742 that leaves substitution undefined when A is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2748, so we will allow direct use of df-sbc 2742. 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 (A = B → ([A / x]φ[B / x]φ))

Proof of Theorem dfsbcq
StepHypRef Expression
1 eleq1 2082 . 2 (A = B → (A {xφ} ↔ B {xφ}))
2 df-sbc 2742 . 2 ([A / x]φA {xφ})
3 df-sbc 2742 . 2 ([B / x]φB {xφ})
41, 2, 33bitr4g 212 1 (A = B → ([A / x]φ[B / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  {cab 2008  [wsbc 2741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018  df-sbc 2742
This theorem is referenced by:  sbceq1d  2746  sbc8g  2748  spsbc  2752  sbcco  2762  sbcco2  2763  sbcie2g  2773  elrabsf  2778  eqsbc3  2779  csbeq1  2832  sbcnestgf  2874  sbcco3g  2880  cbvralcsf  2885  cbvrexcsf  2886  findes  4253  ralrnmpt  5234  rexrnmpt  5235
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