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Theorem dfsbcq 2993
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 2992 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 2994 instead of df-sbc 2992. (dfsbcq2 2994 is needed because unlike Quine we do not overload the df-sb 1630 syntax.) As a consequence of these theorems, we can derive sbc8g 2998, which is a weaker version of df-sbc 2992 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 2998, so we will allow direct use of df-sbc 2992 after theorem sbc2or 2999 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  |-  ( A  =  B  ->  ( [. A  /  x ]. ph  <->  [. B  /  x ]. ph ) )

Proof of Theorem dfsbcq
StepHypRef Expression
1 eleq1 2343 . 2  |-  ( A  =  B  ->  ( A  e.  { x  |  ph }  <->  B  e.  { x  |  ph }
) )
2 df-sbc 2992 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
3 df-sbc 2992 . 2  |-  ( [. B  /  x ]. ph  <->  B  e.  { x  |  ph }
)
41, 2, 33bitr4g 279 1  |-  ( A  =  B  ->  ( [. A  /  x ]. ph  <->  [. B  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991
This theorem is referenced by:  sbceq1d  2996  sbc8g  2998  spsbc  3003  sbcco  3013  sbcco2  3014  sbcie2g  3024  elrabsf  3029  eqsbc3  3030  csbeq1  3084  sbcnestgf  3128  sbcco3g  3136  cbvralcsf  3143  tfindes  4653  findes  4686  ralrnmpt  5669  findcard2  7098  ac6sfi  7101  indexfi  7163  ac6num  8106  fpwwe2cbv  8252  fpwwe2lem2  8254  fpwwe2lem3  8255  nn1suc  9767  uzindOLD  10106  uzind4s  10278  uzind4s2  10279  fzrevral  10866  fzshftral  10869  wrdind  11477  cjth  11588  prmind2  12769  isprs  14064  isdrs  14068  joinlem  14124  meetlem  14131  istos  14141  isdlat  14296  gsumvalx  14451  islmod  15631  elmptrab  17522  isfildlem  17552  quotval  19672  ifeqeqx  23034  sbcbidv2  24969  bisig0  26062  isibg2  26110  sdclem2  26452  fdc  26455  fdc1  26456  sbccomieg  26870  rexrabdioph  26875  rexfrabdioph  26876  2rexfrabdioph  26877  3rexfrabdioph  26878  4rexfrabdioph  26879  6rexfrabdioph  26880  7rexfrabdioph  26881  2nn0ind  27030  zindbi  27031  2sbc6g  27615  2sbc5g  27616  pm14.122b  27623  pm14.24  27632  iotavalsb  27633  sbiota1  27634  iotasbcq  27637  fvsb  27655  bnj609  28949  bnj601  28952  bnj944  28970  lshpkrlem3  29302  hdmap1ffval  31986  hdmap1fval  31987  hdmapffval  32019  hdmapfval  32020  hgmapffval  32078  hgmapfval  32079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279  df-sbc 2992
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