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Theorem necon4ddc 2378
 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 2307 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
3 df-ne 2307 . . . 4 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
42, 3imbi12i 238 . . 3 ((𝐴𝐵𝐶𝐷) ↔ (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
51, 4syl6ib 160 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)))
6 condc 838 . 2 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷) → (𝐶 = 𝐷𝐴 = 𝐵)))
75, 6sylcom 28 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 819   = wceq 1331   ≠ wne 2306 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 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2307 This theorem is referenced by: (None)
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