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Theorem cbvcsbw 3010
Description: Version of cbvcsb 3011 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvcsbw.1  |-  F/_ y C
cbvcsbw.2  |-  F/_ x D
cbvcsbw.3  |-  ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
cbvcsbw  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    C( x, y)    D( x, y)

Proof of Theorem cbvcsbw
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvcsbw.1 . . . . 5  |-  F/_ y C
21nfcri 2276 . . . 4  |-  F/ y  z  e.  C
3 cbvcsbw.2 . . . . 5  |-  F/_ x D
43nfcri 2276 . . . 4  |-  F/ x  z  e.  D
5 cbvcsbw.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2210 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbcw 2939 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2256 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3007 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3007 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2171 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   {cab 2126   F/_wnfc 2269   [.wsbc 2912   [_csb 3006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-sbc 2913  df-csb 3007
This theorem is referenced by:  cbvprod  11358
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