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

Theorem nebidc 2285
Description: Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Assertion
Ref Expression
nebidc (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))))

Proof of Theorem nebidc
StepHypRef Expression
1 id 19 . . . 4 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐵𝐶 = 𝐷))
21necon3bid 2246 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐵𝐶𝐷))
3 id 19 . . . . . . . 8 ((𝐴𝐵𝐶𝐷) → (𝐴𝐵𝐶𝐷))
43a1d 22 . . . . . . 7 ((𝐴𝐵𝐶𝐷) → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷)))
54a1d 22 . . . . . 6 ((𝐴𝐵𝐶𝐷) → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷))))
65necon4biddc 2280 . . . . 5 ((𝐴𝐵𝐶𝐷) → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵𝐶 = 𝐷))))
76com3l 75 . . . 4 (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴𝐵𝐶𝐷) → (𝐴 = 𝐵𝐶 = 𝐷))))
87imp 115 . . 3 ((DECID 𝐴 = 𝐵DECID 𝐶 = 𝐷) → ((𝐴𝐵𝐶𝐷) → (𝐴 = 𝐵𝐶 = 𝐷)))
92, 8impbid2 131 . 2 ((DECID 𝐴 = 𝐵DECID 𝐶 = 𝐷) → ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷)))
109ex 108 1 (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  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-in1 544  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