ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon1addc GIF version

Theorem necon1addc 2331
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1addc.1 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
Assertion
Ref Expression
necon1addc (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))

Proof of Theorem necon1addc
StepHypRef Expression
1 df-ne 2256 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1addc.1 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
3 con1dc 791 . . 3 (DECID 𝜓 → ((¬ 𝜓𝐴 = 𝐵) → (¬ 𝐴 = 𝐵𝜓)))
42, 3sylcom 28 . 2 (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵𝜓)))
51, 4syl7bi 163 1 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 780   = wceq 1289  wne 2255
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator