Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  neeq12d GIF version

Theorem neeq12d 2329
 Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
neeq1d.1 (𝜑𝐴 = 𝐵)
neeq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
neeq12d (𝜑 → (𝐴𝐶𝐵𝐷))

Proof of Theorem neeq12d
StepHypRef Expression
1 neeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21neeq1d 2327 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 neeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43neeq2d 2328 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 187 1 (𝜑 → (𝐴𝐶𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ≠ wne 2309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-ne 2310 This theorem is referenced by:  3netr3d  2341  3netr4d  2342  ennnfonelemim  12007  ctinfom  12011  apdiff  13461
 Copyright terms: Public domain W3C validator