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Theorem mteqand 3045
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 2945 . . 3 (𝜑 → ¬ 𝐶 = 𝐷)
3 mteqand.2 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
42, 3mtand 815 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
54neqned 2947 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wne 2940
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 206  df-an 398  df-ne 2941
This theorem is referenced by:  isdrngd  20226  qsidomlem2  32274  zarcmplem  32519  imadrhmcl  40761  ricdrng1  40762  remul01  40919  remulinvcom  40944  prjspersym  40988  prjspreln0  40990  prjspner1  41007  flt0  41018  fltne  41025
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