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Theorem necon4ddc 2277
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4ddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷)))
Assertion
Ref Expression
necon4ddc (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵)))

Proof of Theorem necon4ddc
StepHypRef Expression
1 necon4ddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷)))
2 df-ne 2206 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
3 df-ne 2206 . . . 4 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
42, 3imbi12i 228 . . 3 ((𝐴𝐵𝐶𝐷) ↔ (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
51, 4syl6ib 150 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)))
6 condc 749 . 2 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷) → (𝐶 = 𝐷𝐴 = 𝐵)))
75, 6sylcom 25 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 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-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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