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Syntax Definition wal 1472
Description: Extend wff definition to include the universal quantifier ('for all'). 𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥." Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
Hypotheses
Ref Expression
wph wff 𝜑
vx.wal setvar 𝑥
Assertion
Ref Expression
wal wff 𝑥𝜑

See definition df-tru 1477 for more information.

Colors of variables: wff setvar class
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