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Theorem csbco 3082
Description: Composition law for chained substitutions into a class.

Use the weaker csbcow 3083 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)

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 3073 . . . . . 6  |-  [_ y  /  x ]_ B  =  { z  |  [. y  /  x ]. z  e.  B }
21abeq2i 2300 . . . . 5  |-  ( z  e.  [_ y  /  x ]_ B  <->  [. y  /  x ]. z  e.  B
)
32sbcbii 3037 . . . 4  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  y ]. [. y  /  x ]. z  e.  B
)
4 sbcco 2999 . . . 4  |-  ( [. A  /  y ]. [. y  /  x ]. z  e.  B  <->  [. A  /  x ]. z  e.  B
)
53, 4bitri 184 . . 3  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  x ]. z  e.  B
)
65abbii 2305 . 2  |-  { z  |  [. A  / 
y ]. z  e.  [_ y  /  x ]_ B }  =  { z  |  [. A  /  x ]. z  e.  B }
7 df-csb 3073 . 2  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  { z  |  [. A  /  y ]. z  e.  [_ y  /  x ]_ B }
8 df-csb 3073 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
96, 7, 83eqtr4i 2220 1  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   {cab 2175   [.wsbc 2977   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  csbvarg  3100  csbnest1g  3127  zsumdc  11433  fsum3  11436  fsumsplitf  11457
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