ForAll[x, expr]
represents the statement that expr is True for all values of .

ForAll[x, cond, expr]
states that expr is True for all x satisfying the condition cond.

ForAll[{x1, x2, ...}, expr]
states that expr is True for all values of all the .


  • ForAll[x, expr] can be entered as . The character can be entered as EscfaEsc or \[ForAll]. The variable is given as a subscript.
  • ForAll[x, cond, expr] can be entered as .
  • In StandardForm, ForAll[x, expr] is output as .
  • ForAll[x, cond, expr] is output as .
  • ForAll can be used in such functions as Reduce, Resolve, and FullSimplify.
  • The condition cond is often used to specify the domain of a variable, as in xIntegers.
  • ForAll[x, cond, expr] is equivalent to ForAll[x, Implies[cond, expr]].
  • ForAll[{x1, x2, ...}, ...] is equivalent to .
  • The value of in ForAll[x, expr] is taken to be localized, as in Block.

ExamplesExamplesopen allclose all

Basic Examples (1)Basic Examples (1)

This states that for all , is positive:

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Use Resolve to get a condition on real parameters for which the statement is true:

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Reduce gives the condition in a solved form:

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