ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon4bddc GIF version

Theorem necon4bddc 2276
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4bddc.1 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))
Assertion
Ref Expression
necon4bddc (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))

Proof of Theorem necon4bddc
StepHypRef Expression
1 necon4bddc.1 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))
2 df-ne 2206 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2syl8ib 155 . 2 (𝜑 → (DECID 𝜓 → (¬ 𝜓 → ¬ 𝐴 = 𝐵)))
4 condc 749 . 2 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝐴 = 𝐵) → (𝐴 = 𝐵𝜓)))
53, 4sylcom 25 1 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 742   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator