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Theorem necon1bidc 2358
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon1bidc.1 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
Assertion
Ref Expression
necon1bidc (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))

Proof of Theorem necon1bidc
StepHypRef Expression
1 df-ne 2307 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bidc.1 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
31, 2syl5bir 152 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝜑))
4 con1dc 841 . 2 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵𝜑) → (¬ 𝜑𝐴 = 𝐵)))
53, 4mpd 13 1 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 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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