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Theorem cbvcsb 3247
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 2565 . . . 4  |-  F/ y  z  e.  C
3 cbvcsb.2 . . . . 5  |-  F/_ x D
43nfcri 2565 . . . 4  |-  F/ x  z  e.  D
5 cbvcsb.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2502 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbc 3181 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2547 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3244 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3244 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2465 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558   [.wsbc 3153   [_csb 3243
This theorem is referenced by:  cbvcsbv  3248  cbvsum  12481  measiuns  24563  cbvprod  25233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-sbc 3154  df-csb 3244
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