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Theorem cbvcsb 3191
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 2510 . . . 4  |-  F/ y  z  e.  C
3 cbvcsb.2 . . . . 5  |-  F/_ x D
43nfcri 2510 . . . 4  |-  F/ x  z  e.  D
5 cbvcsb.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2447 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbc 3125 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2492 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3188 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3188 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2410 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {cab 2366   F/_wnfc 2503   [.wsbc 3097   [_csb 3187
This theorem is referenced by:  cbvcsbv  3192  cbvsum  12409  measiuns  24358  cbvprod  25013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-sbc 3098  df-csb 3188
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