ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon4addc GIF version

Theorem necon4addc 2325
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4addc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜓)))
Assertion
Ref Expression
necon4addc (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵)))

Proof of Theorem necon4addc
StepHypRef Expression
1 necon4addc.1 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜓)))
2 df-ne 2256 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32imbi1i 236 . . 3 ((𝐴𝐵 → ¬ 𝜓) ↔ (¬ 𝐴 = 𝐵 → ¬ 𝜓))
4 condc 787 . . 3 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → ¬ 𝜓) → (𝜓𝐴 = 𝐵)))
53, 4syl5bi 150 . 2 (DECID 𝐴 = 𝐵 → ((𝐴𝐵 → ¬ 𝜓) → (𝜓𝐴 = 𝐵)))
61, 5sylcom 28 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵)))
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-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