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

Theorem csbnestgf 3022
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 2671 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 df-csb 2976 . . . . . . 7  |-  [_ B  /  y ]_ C  =  { z  |  [. B  /  y ]. z  e.  C }
32abeq2i 2228 . . . . . 6  |-  ( z  e.  [_ B  / 
y ]_ C  <->  [. B  / 
y ]. z  e.  C
)
43sbcbii 2940 . . . . 5  |-  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. A  /  x ]. [. B  /  y ]. z  e.  C
)
5 nfcr 2250 . . . . . . 7  |-  ( F/_ x C  ->  F/ x  z  e.  C )
65alimi 1416 . . . . . 6  |-  ( A. y F/_ x C  ->  A. y F/ x  z  e.  C )
7 sbcnestgf 3021 . . . . . 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 284 . . . . 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 2235 . . 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 281 . 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 2976 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  { z  |  [. A  /  x ]. z  e.  [_ B  /  y ]_ C }
13 df-csb 2976 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ C  =  {
z  |  [. [_ A  /  x ]_ B  / 
y ]. z  e.  C }
1411, 12, 133eqtr4g 2175 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 1314    = wceq 1316   F/wnf 1421    e. wcel 1465   {cab 2103   F/_wnfc 2245   _Vcvv 2660   [.wsbc 2882   [_csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbnestg  3024  csbnest1g  3025
  Copyright terms: Public domain W3C validator