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Theorem mteqand 3114
 Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.)
Hypotheses
Ref Expression
mteqand.1 (𝜑𝐶𝐷)
mteqand.2 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mteqand (𝜑𝐴𝐵)

Proof of Theorem mteqand
StepHypRef Expression
1 mteqand.1 . . . 4 (𝜑𝐶𝐷)
21neneqd 3016 . . 3 (𝜑 → ¬ 𝐶 = 𝐷)
3 mteqand.2 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
42, 3mtand 815 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
54neqned 3018 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 3011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-ne 3012 This theorem is referenced by:  qsidomlem2  31008  uvcn0  39402  remul01  39489  remulinvcom  39512  prjspersym  39531  prjspreln0  39533  fltne  39546
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