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Theorem sbcne12 3827
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcne12

Proof of Theorem sbcne12
StepHypRef Expression
1 nne 2658 . . . . . 6
21sbcbii 3387 . . . . 5
32a1i 11 . . . 4
4 sbcng 3368 . . . 4
5 sbceqg 3825 . . . . 5
6 nne 2658 . . . . 5
75, 6syl6bbr 263 . . . 4
83, 4, 73bitr3d 283 . . 3
98con4bid 293 . 2
10 sbcex 3337 . . . 4
1110con3i 135 . . 3
12 csbprc 3821 . . . . 5
13 csbprc 3821 . . . . 5
1412, 13eqtr4d 2501 . . . 4
1514, 6sylibr 212 . . 3
1611, 152falsed 351 . 2
179, 16pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  e.wcel 1818  =/=wne 2652   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784
This theorem is referenced by:  disjdsct  27521  cdlemkid3N  36659  cdlemkid4  36660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785
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