ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-4 Unicode version

Axiom ax-4 1472
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.) 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 1410. Conditional forms of the converse are given by ax-12 1474, ax-16 1770, and ax-17 1491.

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 1733.

(Contributed by NM, 5-Aug-1993.)

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  setvar  x
31, 2wal 1314 . 2  wff  A. x ph
43, 1wi 4 1  wff  ( A. x ph  ->  ph )
Colors of variables: wff set class
This axiom is referenced by:  sp  1473  ax-12  1474  hbequid  1478  spi  1501  hbim  1509  19.3h  1517  19.21h  1521  19.21bi  1522  hbimd  1537  19.21ht  1545  hbnt  1616  19.12  1628  19.38  1639  ax9o  1661  hbae  1681  equveli  1717  sb2  1725  drex1  1754  ax11b  1782  a16gb  1821  sb56  1841  sb6  1842  sbalyz  1952  hbsb4t  1966  moim  2041  mopick  2055
  Copyright terms: Public domain W3C validator