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Theorem csbnestgf 3018
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 2668 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 df-csb 2972 . . . . . . 7  |-  [_ B  /  y ]_ C  =  { z  |  [. B  /  y ]. z  e.  C }
32abeq2i 2225 . . . . . 6  |-  ( z  e.  [_ B  / 
y ]_ C  <->  [. B  / 
y ]. z  e.  C
)
43sbcbii 2936 . . . . 5  |-  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. A  /  x ]. [. B  /  y ]. z  e.  C
)
5 nfcr 2247 . . . . . . 7  |-  ( F/_ x C  ->  F/ x  z  e.  C )
65alimi 1414 . . . . . 6  |-  ( A. y F/_ x C  ->  A. y F/ x  z  e.  C )
7 sbcnestgf 3017 . . . . . 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 282 . . . . 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 191 . . . 4  |-  ( ( A  e.  _V  /\  A. y F/_ x C )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. [_ A  /  x ]_ B  /  y ]. z  e.  C
) )
109abbidv 2232 . . 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 279 . 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 2972 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  { z  |  [. A  /  x ]. z  e.  [_ B  /  y ]_ C }
13 df-csb 2972 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ C  =  {
z  |  [. [_ A  /  x ]_ B  / 
y ]. z  e.  C }
1411, 12, 133eqtr4g 2172 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    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314   F/wnf 1419    e. wcel 1463   {cab 2101   F/_wnfc 2242   _Vcvv 2657   [.wsbc 2878   [_csb 2971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-csb 2972
This theorem is referenced by:  csbnestg  3020  csbnest1g  3021
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