Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcne12gOLD Structured version   Unicode version

Theorem sbcne12gOLD 3823
 Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcne12 3822 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcne12gOLD

Proof of Theorem sbcne12gOLD
StepHypRef Expression
1 nne 2663 . . . . 5
21sbcbii 3386 . . . 4
32a1i 11 . . 3
4 sbcng 3367 . . 3
5 sbceqg 3820 . . . 4
6 nne 2663 . . . 4
75, 6syl6bbr 263 . . 3
83, 4, 73bitr3d 283 . 2
98con4bid 293 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wceq 1374   wcel 1762   wne 2657  wsbc 3326  csb 3430 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-v 3110  df-sbc 3327  df-csb 3431 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator