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Theorem sbcne12 3774
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 2627 . . . . . 6  |-  ( -.  B  =/=  C  <->  B  =  C )
21sbcbii 3322 . . . . 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 3307 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  -. 
[. A  /  x ]. B  =/=  C
) )
5 sbceqg 3772 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
6 nne 2627 . . . . 5  |-  ( -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
)
75, 6syl6bbr 267 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
83, 4, 73bitr3d 287 . . 3  |-  ( A  e.  _V  ->  ( -.  [. A  /  x ]. B  =/=  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
98con4bid 295 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
10 sbcex 3276 . . . 4  |-  ( [. A  /  x ]. B  =/=  C  ->  A  e.  _V )
1110con3i 141 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  =/=  C
)
12 csbprc 3769 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
13 csbprc 3769 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
1412, 13eqtr4d 2487 . . . 4  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
1514, 6sylibr 216 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
1611, 152falsed 353 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
179, 16pm2.61i 168 1  |-  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    = wceq 1443    e. wcel 1886    =/= wne 2621   _Vcvv 3044   [.wsbc 3266   [_csb 3362   (/)c0 3730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731
This theorem is referenced by:  disjdsct  28276  cdlemkid3N  34494  cdlemkid4  34495
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