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Theorem csbco 2889
Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbco  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B
Distinct variable group:    y, B
Allowed substitution hints:    A( x, y)    B( x)

Proof of Theorem csbco
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-csb 2881 . . . . . 6  |-  [_ y  /  x ]_ B  =  { z  |  [. y  /  x ]. z  e.  B }
21abeq2i 2164 . . . . 5  |-  ( z  e.  [_ y  /  x ]_ B  <->  [. y  /  x ]. z  e.  B
)
32sbcbii 2845 . . . 4  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  y ]. [. y  /  x ]. z  e.  B
)
4 sbcco 2808 . . . 4  |-  ( [. A  /  y ]. [. y  /  x ]. z  e.  B  <->  [. A  /  x ]. z  e.  B
)
53, 4bitri 177 . . 3  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  x ]. z  e.  B
)
65abbii 2169 . 2  |-  { z  |  [. A  / 
y ]. z  e.  [_ y  /  x ]_ B }  =  { z  |  [. A  /  x ]. z  e.  B }
7 df-csb 2881 . 2  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  { z  |  [. A  /  y ]. z  e.  [_ y  /  x ]_ B }
8 df-csb 2881 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
96, 7, 83eqtr4i 2086 1  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   {cab 2042   [.wsbc 2787   [_csb 2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881
This theorem is referenced by:  csbvarg  2905  csbnest1g  2929
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