MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbco Unicode version

Theorem csbco 3103
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 3095 . . . . . 6  |-  [_ y  /  x ]_ B  =  { z  |  [. y  /  x ]. z  e.  B }
21abeq2i 2403 . . . . 5  |-  ( z  e.  [_ y  /  x ]_ B  <->  [. y  /  x ]. z  e.  B
)
32sbcbii 3059 . . . 4  |-  ( [. A  /  y ]. z  e.  [_ y  /  x ]_ B  <->  [. A  /  y ]. [. y  /  x ]. z  e.  B
)
4 sbcco 3026 . . . 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 2408 . 2  |-  { z  |  [. A  / 
y ]. z  e.  [_ y  /  x ]_ B }  =  { z  |  [. A  /  x ]. z  e.  B }
7 df-csb 3095 . 2  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  { z  |  [. A  /  y ]. z  e.  [_ y  /  x ]_ B }
8 df-csb 3095 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
96, 7, 83eqtr4i 2326 1  |-  [_ A  /  y ]_ [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  csbvarg  3121  csbnest1g  3146  zsum  12207  fsum  12209  zprod  24160  fprod  24164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
  Copyright terms: Public domain W3C validator