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Theorem necon1aidc 2234
 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 2188 . 2 (AB ↔ ¬ A = B)
2 necon1aidc.1 . . 3 (DECID φ → (¬ φA = B))
3 con1dc 746 . . 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 733   = wceq 1228   ≠ wne 2186 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 532  ax-in2 533  ax-io 617 This theorem depends on definitions:  df-bi 110  df-dc 734  df-ne 2188 This theorem is referenced by:  necon1idc  2236
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