MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-4 Unicode version

Axiom ax-4 2079
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all  x, it is true for any specific  x (that would typically occur as a free variable in the wff substituted for  ph). (A free variable is one that does not occur in the scope of a quantifier:  x and  y are both free in  x  =  y, but only  x is free in  A. y x  =  y.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2079 through ax-7 1711 plus rule ax-gen 1535). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1535. Conditional forms of the converse are given by ax-12 1870, ax-15 2087, ax-16 2088, and ax-17 1605.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from  x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1971.

An interesting alternate axiomatization uses ax467 2112 and ax-5o 2080 in place of ax-4 2079, ax-5 1546, ax-6 1706, and ax-7 1711.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1719. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-4  |-  ( A. x ph  ->  ph )

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
31, 2wal 1529 . 2  wff  A. x ph
43, 1wi 6 1  wff  ( A. x ph  ->  ph )
Colors of variables: wff set class
This axiom is referenced by:  ax5  2089  ax6  2090  hba1-o  2092  hbae-o  2096  ax11  2098  ax12  2099  equid1  2101  sps-o  2102  ax46  2105  ax67to6  2110  ax467  2112  ax11indalem  2139  ax11inda2ALT  2140
  Copyright terms: Public domain W3C validator