ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon3ad GIF version

Theorem necon3ad 2325
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ad.1 (𝜑 → (𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon3ad (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Proof of Theorem necon3ad
StepHypRef Expression
1 df-ne 2284 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3ad.1 . . 3 (𝜑 → (𝜓𝐴 = 𝐵))
32con3d 603 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓))
41, 3syl5bi 151 1 (𝜑 → (𝐴𝐵 → ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1314  wne 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 586  ax-in2 587
This theorem depends on definitions:  df-bi 116  df-ne 2284
This theorem is referenced by:  necon3d  2327  disjnim  3888  fodjumkvlemres  6999  nlt1pig  7113  eucalglt  11634  nprm  11700  0nnei  12217
  Copyright terms: Public domain W3C validator