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Theorem cbvcsb 3098
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 2426 . . . 4  |-  F/ y  z  e.  C
3 cbvcsb.2 . . . . 5  |-  F/_ x D
43nfcri 2426 . . . 4  |-  F/ x  z  e.  D
5 cbvcsb.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2363 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbc 3032 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2408 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3095 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3095 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2326 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  cbvcsbv  3099  measiuns  23559  cbvcprod  24137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-sbc 3005  df-csb 3095
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