ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcnestgf Structured version   Unicode version

Theorem sbcnestgf 2891
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf  V  F/  [.  ]. [.  ].  [. [_  ]_  ].

Proof of Theorem sbcnestgf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2760 . . . . 5  [.  ].
[.  ].  [.  ]. [.  ].
2 csbeq1 2849 . . . . . 6  [_  ]_  [_  ]_
3 dfsbcq 2760 . . . . . 6  [_  ]_  [_  ]_  [. [_  ]_  ].  [. [_  ]_  ].
42, 3syl 14 . . . . 5  [. [_  ]_  ].  [.
[_  ]_  ].
51, 4bibi12d 224 . . . 4  [.  ].
[.  ].  [. [_  ]_  ].  [.  ]. [.  ].  [. [_  ]_  ].
65imbi2d 219 . . 3  F/  [.  ]. [.  ].  [. [_  ]_  ].  F/  [.  ].
[.  ].  [. [_  ]_  ].
7 vex 2554 . . . . 5 
_V
87a1i 9 . . . 4  F/  _V
9 csbeq1a 2854 . . . . . 6  [_  ]_
10 dfsbcq 2760 . . . . . 6  [_  ]_  [.  ].  [. [_  ]_  ].
119, 10syl 14 . . . . 5  [.  ].  [. [_  ]_  ].
1211adantl 262 . . . 4  F/  [.  ].  [. [_  ]_  ].
13 nfnf1 1433 . . . . 5  F/ F/
1413nfal 1465 . . . 4  F/ F/
15 nfa1 1431 . . . . 5  F/ F/
16 nfcsb1v 2876 . . . . . 6  F/_ [_  ]_
1716a1i 9 . . . . 5  F/  F/_ [_  ]_
18 sp 1398 . . . . 5  F/  F/
1915, 17, 18nfsbcd 2777 . . . 4  F/  F/ [. [_  ]_  ].
208, 12, 14, 19sbciedf 2792 . . 3  F/  [.  ].
[.  ].  [. [_  ]_  ].
216, 20vtoclg 2607 . 2  V  F/  [.  ]. [.  ].  [. [_  ]_  ].
2221imp 115 1  V  F/  [.  ]. [.  ].  [. [_  ]_  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   F/wnf 1346   wcel 1390   F/_wnfc 2162   _Vcvv 2551   [.wsbc 2758   [_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:  csbnestgf  2892  sbcnestg  2893
  Copyright terms: Public domain W3C validator