ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcne12g Unicode version

Theorem sbcne12g 2947
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcne12g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )

Proof of Theorem sbcne12g
StepHypRef Expression
1 sbceqg 2945 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
21notbid 627 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
3 df-ne 2256 . . . . 5  |-  ( B  =/=  C  <->  -.  B  =  C )
43sbcbii 2896 . . . 4  |-  ( [. A  /  x ]. B  =/=  C  <->  [. A  /  x ].  -.  B  =  C )
5 sbcng 2877 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  =  C  <->  -.  [. A  /  x ]. B  =  C
) )
64, 5syl5bb 190 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  -. 
[. A  /  x ]. B  =  C
) )
7 df-ne 2256 . . . 4  |-  ( [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
)
87a1i 9 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  = 
[_ A  /  x ]_ C ) )
96, 8bibi12d 233 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )  <->  ( -.  [. A  /  x ]. B  =  C  <->  -.  [_ A  /  x ]_ B  = 
[_ A  /  x ]_ C ) ) )
102, 9mpbird 165 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438    =/= wne 2255   [.wsbc 2838   [_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-in1 579  ax-in2 580  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-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator