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Theorem sbcne12 3800
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 2622 . . . . . 6  |-  ( -.  B  =/=  C  <->  B  =  C )
21sbcbii 3352 . . . . 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 3337 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  -. 
[. A  /  x ]. B  =/=  C
) )
5 sbceqg 3798 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
6 nne 2622 . . . . 5  |-  ( -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
)
75, 6syl6bbr 266 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
83, 4, 73bitr3d 286 . . 3  |-  ( A  e.  _V  ->  ( -.  [. A  /  x ]. B  =/=  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
98con4bid 294 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
10 sbcex 3306 . . . 4  |-  ( [. A  /  x ]. B  =/=  C  ->  A  e.  _V )
1110con3i 140 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  =/=  C
)
12 csbprc 3795 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
13 csbprc 3795 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
1412, 13eqtr4d 2464 . . . 4  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
1514, 6sylibr 215 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
1611, 152falsed 352 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
179, 16pm2.61i 167 1  |-  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   [.wsbc 3296   [_csb 3392   (/)c0 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-in 3440  df-ss 3447  df-nul 3759
This theorem is referenced by:  disjdsct  28120  cdlemkid3N  34209  cdlemkid4  34210
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