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Theorem cbvcsbw 3053
Description: Version of cbvcsb 3054 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 2306 . . . 4  |-  F/ y  z  e.  C
3 cbvcsbw.2 . . . . 5  |-  F/_ x D
43nfcri 2306 . . . 4  |-  F/ x  z  e.  D
5 cbvcsbw.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2240 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbcw 2982 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2286 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3050 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3050 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2201 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   {cab 2156   F/_wnfc 2299   [.wsbc 2955   [_csb 3049
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050
This theorem is referenced by:  cbvprod  11508
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