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Theorem sbcne12g 2991
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 2989 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
21notbid 641 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
3 df-ne 2286 . . . . 5  |-  ( B  =/=  C  <->  -.  B  =  C )
43sbcbii 2940 . . . 4  |-  ( [. A  /  x ]. B  =/=  C  <->  [. A  /  x ].  -.  B  =  C )
5 sbcng 2921 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  =  C  <->  -.  [. A  /  x ]. B  =  C
) )
64, 5syl5bb 191 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  -. 
[. A  /  x ]. B  =  C
) )
7 df-ne 2286 . . . 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 234 . 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 166 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 104    = wceq 1316    e. wcel 1465    =/= wne 2285   [.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-in1 588  ax-in2 589  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-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by: (None)
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