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Theorem cbvcsb 2975
Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on  A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvcsb.1  |-  F/_ y C
cbvcsb.2  |-  F/_ x D
cbvcsb.3  |-  ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
cbvcsb  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D

Proof of Theorem cbvcsb
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvcsb.1 . . . . 5  |-  F/_ y C
21nfcri 2249 . . . 4  |-  F/ y  z  e.  C
3 cbvcsb.2 . . . . 5  |-  F/_ x D
43nfcri 2249 . . . 4  |-  F/ x  z  e.  D
5 cbvcsb.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2184 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbc 2905 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2230 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 2972 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 2972 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2145 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   {cab 2101   F/_wnfc 2242   [.wsbc 2878   [_csb 2971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-sbc 2879  df-csb 2972
This theorem is referenced by:  cbvcsbv  2976  cbvsum  11021
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