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Syntax Definition wal 1266
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 set x
Assertion
Ref Expression
wal wff xφ

This syntax is primitive. The first axiom using it is ax-5 1267.

Colors of variables: wff set class
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