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Theorem csbnestgf 3142
Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
Assertion
Ref Expression
csbnestgf  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)

Proof of Theorem csbnestgf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 df-csb 3095 . . . . . . 7  |-  [_ B  /  y ]_ C  =  { z  |  [. B  /  y ]. z  e.  C }
32abeq2i 2403 . . . . . 6  |-  ( z  e.  [_ B  / 
y ]_ C  <->  [. B  / 
y ]. z  e.  C
)
43sbcbii 3059 . . . . 5  |-  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. A  /  x ]. [. B  /  y ]. z  e.  C
)
5 nfcr 2424 . . . . . . 7  |-  ( F/_ x C  ->  F/ x  z  e.  C )
65alimi 1549 . . . . . 6  |-  ( A. y F/_ x C  ->  A. y F/ x  z  e.  C )
7 sbcnestgf 3141 . . . . . 6  |-  ( ( A  e.  _V  /\  A. y F/ x  z  e.  C )  -> 
( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [.
[_ A  /  x ]_ B  /  y ]. z  e.  C
) )
86, 7sylan2 460 . . . . 5  |-  ( ( A  e.  _V  /\  A. y F/_ x C )  ->  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. [_ A  /  x ]_ B  /  y ]. z  e.  C
) )
94, 8syl5bb 248 . . . 4  |-  ( ( A  e.  _V  /\  A. y F/_ x C )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. [_ A  /  x ]_ B  /  y ]. z  e.  C
) )
109abbidv 2410 . . 3  |-  ( ( A  e.  _V  /\  A. y F/_ x C )  ->  { z  |  [. A  /  x ]. z  e.  [_ B  /  y ]_ C }  =  { z  |  [. [_ A  /  x ]_ B  /  y ]. z  e.  C } )
111, 10sylan 457 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  { z  |  [. A  /  x ]. z  e.  [_ B  /  y ]_ C }  =  { z  |  [. [_ A  /  x ]_ B  /  y ]. z  e.  C } )
12 df-csb 3095 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  { z  |  [. A  /  x ]. z  e.  [_ B  /  y ]_ C }
13 df-csb 3095 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ C  =  {
z  |  [. [_ A  /  x ]_ B  / 
y ]. z  e.  C }
1411, 12, 133eqtr4g 2353 1  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   _Vcvv 2801   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  csbnestg  3144  csbnest1g  3146
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-3an 936  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
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