@ChrisK That is expected, isn't it? p = False, then p⇒q is true, regardless if q is Ture or False. Am I wrog here?

Simplify@@@Tuples[{True,False},2]

You do get the General::fas warning, "This message is generated when the symbol False is encountered in a position where valid assumptions are expected."

Column[Row[{"(",#2,"\[Implies]",#1, ") is ", Simplify[#1,#2]}]&@@@Tuples[{True,False},2]]

en.wikipedia.org/wiki/… Column 11 :)

@CarlLange Exactly. @ChrisK Can you explain why is your expectation for Simplify different?

Going even further, if there is a problem is that Simplify[undefined,False] doesn't evaluate to True as it should.

@ChrisK Only now I noticed you did ask the question, so I explained my comment on this answer.

@rhermans Thanks! My problem was that forgotten $Assumptions could lead to confusion later on. The actual case was setting $Assumptions={x>0} early on, then setting x=0 and breaking Simplify's expected behavior. Especially dangerous for me, because both parts happen behind the scenes of my package, so users would have no idea what's going on. But I figured out how to fix it for my purposes.

@ChrisK Probably best to limit the scope of $Assumptions with Assuming or Block.

@rhermans Good advice!

(Shameless self-plug) In case you ever needed a 3D version of Locator to position stuff interactively in 3D, there is now ResourceFunction["Locator3D"]:

DynamicModule[{p1 = {0, 0, 0}, p2 = {0, 0, 1}, p3 = {1, 0, 0}},
 Graphics3D[
  {
   ResourceFunction["Locator3D"][Dynamic@p1],
   ResourceFunction["Locator3D"][Dynamic@p2],
   ResourceFunction["Locator3D"][Dynamic@p3],
   Dynamic@Tube@{p1, p2, p3},
   ResourceFunction["Locator3D"]["AllowedActions" -> All,
    "RotateGuides" -> True]
   },
  PlotRange -> 2, SphericalRegion -> True
  ]
 ]