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Theorem necon4idc 2324
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon4idc.1 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷))
Assertion
Ref Expression
necon4idc (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵))

Proof of Theorem necon4idc
StepHypRef Expression
1 necon4idc.1 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷))
2 df-ne 2256 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
31, 2syl6ib 159 . 2 (DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝐶 = 𝐷))
43necon4aidc 2323 1 (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 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)
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