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Theorem csbnest1g 2895
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  V  [_  ]_ [_  ]_ C 
[_ [_  ]_  ]_ C

Proof of Theorem csbnest1g
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 2876 . . . 4  F/_ [_  ]_ C
21ax-gen 1335 . . 3  F/_ [_  ]_ C
3 csbnestgf 2892 . . 3  V  F/_ [_  ]_ C  [_  ]_ [_  ]_ [_  ]_ C  [_ [_  ]_  ]_ [_  ]_ C
42, 3mpan2 401 . 2  V  [_  ]_ [_  ]_ [_  ]_ C 
[_ [_  ]_  ]_ [_  ]_ C
5 csbco 2855 . . 3  [_  ]_ [_  ]_ C 
[_  ]_ C
65csbeq2i 2870 . 2  [_  ]_ [_  ]_ [_  ]_ C 
[_  ]_
[_  ]_ C
7 csbco 2855 . 2  [_ [_  ]_  ]_ [_  ]_ C  [_ [_  ]_  ]_ C
84, 6, 73eqtr3g 2092 1  V  [_  ]_ [_  ]_ C 
[_ [_  ]_  ]_ C
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   wceq 1242   wcel 1390   F/_wnfc 2162   [_csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  csbidmg  2896
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