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Theorem sbcne12 3780
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcne12  |-  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )

Proof of Theorem sbcne12
StepHypRef Expression
1 nne 2650 . . . . . 6  |-  ( -.  B  =/=  C  <->  B  =  C )
21sbcbii 3347 . . . . 5  |-  ( [. A  /  x ].  -.  B  =/=  C  <->  [. A  /  x ]. B  =  C )
32a1i 11 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  [. A  /  x ]. B  =  C )
)
4 sbcng 3328 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  -. 
[. A  /  x ]. B  =/=  C
) )
5 sbceqg 3778 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
6 nne 2650 . . . . 5  |-  ( -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
)
75, 6syl6bbr 263 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
83, 4, 73bitr3d 283 . . 3  |-  ( A  e.  _V  ->  ( -.  [. A  /  x ]. B  =/=  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
98con4bid 293 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
10 sbcex 3297 . . . 4  |-  ( [. A  /  x ]. B  =/=  C  ->  A  e.  _V )
1110con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  =/=  C
)
12 csbprc 3774 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
13 csbprc 3774 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
1412, 13eqtr4d 2495 . . . 4  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
1514, 6sylibr 212 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
1611, 152falsed 351 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
179, 16pm2.61i 164 1  |-  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071   [.wsbc 3287   [_csb 3389   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739
This theorem is referenced by:  disjdsct  26142  cdlemkid3N  34886  cdlemkid4  34887
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