Metamath Proof Explorer

Syntax definition wal

Description: Extend wff definition to include the universal quantifier ("for all"). A. x ph 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.

Ref Expression
Assertion wal wff x φ