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Axiom ax-4 1443
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 1381. Conditional forms of the converse are given by ax-12 1445, ax-16 1739, and ax-17 1462.

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 1702.

(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 1285 . 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  1444  ax-12  1445  hbequid  1449  spi  1472  hbim  1480  19.3h  1488  19.21h  1492  19.21bi  1493  hbimd  1508  19.21ht  1516  hbnt  1586  19.12  1598  19.38  1609  ax9o  1631  hbae  1650  equveli  1686  sb2  1694  drex1  1723  ax11b  1751  a16gb  1790  sb56  1810  sb6  1811  sbalyz  1920  hbsb4t  1934  moim  2009  mopick  2023
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