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Theorem csbnest1g 2981
Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
csbnest1g  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )

Proof of Theorem csbnest1g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 2961 . . . 4  |-  F/_ x [_ y  /  x ]_ C
21ax-gen 1383 . . 3  |-  A. y F/_ x [_ y  /  x ]_ C
3 csbnestgf 2978 . . 3  |-  ( ( A  e.  V  /\  A. y F/_ x [_ y  /  x ]_ C
)  ->  [_ A  /  x ]_ [_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
42, 3mpan2 416 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
5 csbco 2940 . . 3  |-  [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ B  /  x ]_ C
65csbeq2i 2955 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ A  /  x ]_ [_ B  /  x ]_ C
7 csbco 2940 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C
84, 6, 73eqtr3g 2143 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    = wceq 1289    e. wcel 1438   F/_wnfc 2215   [_csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by:  csbidmg  2982
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