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Theorem necon1idc 2359
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1idc.1 (𝐴𝐵𝐶 = 𝐷)
Assertion
Ref Expression
necon1idc (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵))

Proof of Theorem necon1idc
StepHypRef Expression
1 df-ne 2307 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1idc.1 . . . 4 (𝐴𝐵𝐶 = 𝐷)
31, 2sylbir 134 . . 3 𝐴 = 𝐵𝐶 = 𝐷)
43a1i 9 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝐶 = 𝐷))
54necon1aidc 2357 1 (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 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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