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Axiom ax-4 1416
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 1354. Conditional forms of the converse are given by ax-12 1418, ax-16 1711, and ax-17 1435.

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 1674.

(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 1257 . 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  1417  ax-12  1418  hbequid  1422  spi  1445  hbim  1453  19.3h  1461  19.21h  1465  19.21bi  1466  hbimd  1481  19.21ht  1489  hbnt  1559  19.12  1571  19.38  1582  ax9o  1604  hbae  1622  equveli  1658  sb2  1666  drex1  1695  ax11b  1723  a16gb  1761  sb56  1781  sb6  1782  sbalyz  1891  hbsb4t  1905  moim  1980  mopick  1994
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