MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wal Unicode version

Syntax Definition wal 1532
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
wph  wff  ph
vx  set  x
Ref Expression
wal  wff  A. x ph

This syntax is primitive. The first axiom using it is ax-5 1533.

Colors of variables: wff set class
  Copyright terms: Public domain W3C validator