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Theorem necon1ddc 2402
 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 2325 . 2 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
2 necon1ddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))
32necon1bddc 2401 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷𝐴 = 𝐵)))
41, 3syl7bi 164 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 820   = wceq 1332   ≠ wne 2324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-ne 2325 This theorem is referenced by:  xblss2ps  12743  xblss2  12744
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