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Theorem necon1ddc 2283
 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 2206 . 2 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
2 necon1ddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))
32necon1bddc 2282 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷𝐴 = 𝐵)))
41, 3syl7bi 154 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  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)
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