Syntax Definition wal 1584

Description: Extend wff definition to include the universal quantifier ('for all'). A.xph is read "ph (phi) is true for all x." Typically, in its final application ph 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 ph
vx	set x

Assertion

Ref	Expression
wal	wff A.xph

This syntax is primitive. The first axiom using it is ax-5 1590.

Copyright terms: Public domain