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Theorem cbvcsb 3085
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 2413 . . . 4  |-  F/ y  z  e.  C
3 cbvcsb.2 . . . . 5  |-  F/_ x D
43nfcri 2413 . . . 4  |-  F/ x  z  e.  D
5 cbvcsb.3 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65eleq2d 2350 . . . 4  |-  ( x  =  y  ->  (
z  e.  C  <->  z  e.  D ) )
72, 4, 6cbvsbc 3019 . . 3  |-  ( [. A  /  x ]. z  e.  C  <->  [. A  /  y ]. z  e.  D
)
87abbii 2395 . 2  |-  { z  |  [. A  /  x ]. z  e.  C }  =  { z  |  [. A  /  y ]. z  e.  D }
9 df-csb 3082 . 2  |-  [_ A  /  x ]_ C  =  { z  |  [. A  /  x ]. z  e.  C }
10 df-csb 3082 . 2  |-  [_ A  /  y ]_ D  =  { z  |  [. A  /  y ]. z  e.  D }
118, 9, 103eqtr4i 2313 1  |-  [_ A  /  x ]_ C  = 
[_ A  /  y ]_ D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  cbvcsbv  3086  measiuns  23544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-sbc 2992  df-csb 3082
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