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Theorem necon1ddc 2333
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1ddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))
Assertion
Ref Expression
necon1ddc (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵)))

Proof of Theorem necon1ddc
StepHypRef Expression
1 df-ne 2256 . 2 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
2 necon1ddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))
32necon1bddc 2332 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷𝐴 = 𝐵)))
41, 3syl7bi 163 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 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-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)
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