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Theorem necon1aidc 2256
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon1aidc.1 (DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))
Assertion
Ref Expression
necon1aidc (DECID 𝜑 → (𝐴𝐵𝜑))

Proof of Theorem necon1aidc
StepHypRef Expression
1 df-ne 2206 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1aidc.1 . . 3 (DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))
3 con1dc 753 . . 3 (DECID 𝜑 → ((¬ 𝜑𝐴 = 𝐵) → (¬ 𝐴 = 𝐵𝜑)))
42, 3mpd 13 . 2 (DECID 𝜑 → (¬ 𝐴 = 𝐵𝜑))
51, 4syl5bi 141 1 (DECID 𝜑 → (𝐴𝐵𝜑))
 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:  necon1idc  2258
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