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Theorem csbco 3090
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 3082 . . . . . 6  |-  [_ y  /  x ]_ B  =  { z  |  [. y  /  x ]. z  e.  B }
21abeq2i 2390 . . . . 5  |-  ( z  e.  [_ y  /  x ]_ B  <->  [. y  /  x ]. z  e.  B
)
32sbcbii 3046 . . . 4  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  y ]. [. y  /  x ]. z  e.  B
)
4 sbcco 3013 . . . 4  |-  ( [. A  /  y ]. [. y  /  x ]. z  e.  B  <->  [. A  /  x ]. z  e.  B
)
53, 4bitri 240 . . 3  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  x ]. z  e.  B
)
65abbii 2395 . 2  |-  { z  |  [. A  / 
y ]. z  e.  [_ y  /  x ]_ B }  =  { z  |  [. A  /  x ]. z  e.  B }
7 df-csb 3082 . 2  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  { z  |  [. A  /  y ]. z  e.  [_ y  /  x ]_ B }
8 df-csb 3082 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
96, 7, 83eqtr4i 2313 1  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbvarg  3108  csbnest1g  3133  zsum  12191  fsum  12193
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-v 2790  df-sbc 2992  df-csb 3082
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