ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2ad GIF version

Theorem necon2ad 2308
Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2ad.1 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
Assertion
Ref Expression
necon2ad (𝜑 → (𝜓𝐴𝐵))

Proof of Theorem necon2ad
StepHypRef Expression
1 necon2ad.1 . . 3 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
21con2d 587 . 2 (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
3 df-ne 2252 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 160 1 (𝜑 → (𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1287  wne 2251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2252
This theorem is referenced by:  necon2d  2310  prneimg  3595  tz7.2  4148  nordeq  4326  pr2ne  6741  ltne  7491  apne  8018  xrltne  9187  ge0nemnf  9195  rpexp  10926  sqrt2irr  10935
  Copyright terms: Public domain W3C validator