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Theorem necon4biddc 2330
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4biddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷))))
Assertion
Ref Expression
necon4biddc (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵𝐶 = 𝐷))))

Proof of Theorem necon4biddc
StepHypRef Expression
1 necon4biddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷))))
2 df-ne 2256 . . . 4 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
32bibi2i 225 . . 3 ((𝐴𝐵𝐶𝐷) ↔ (𝐴𝐵 ↔ ¬ 𝐶 = 𝐷))
41, 3syl8ib 164 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵 ↔ ¬ 𝐶 = 𝐷))))
54necon4abiddc 2328 1 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵𝐶 = 𝐷))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  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:  nebidc  2335
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