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Axiom ax-4 1377
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 , it is true for any specific (that would typically occur as a free variable in the wff substituted for ). (A free variable is one that does not occur in the scope of a quantifier: and are both free in , but only is free in .) 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 1314. Conditional forms of the converse are given by ax-12 1379, ax-16 1673, and ax-17 1396.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 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 1636.

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

Ref Expression

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3
2 vx . . 3  setvar
31, 2wal 1224 . 2
43, 1wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  sp  1378  ax-12  1379  hbequid  1383  spi  1407  hbim  1415  19.3h  1423  19.21h  1427  19.21bi  1428  hbimd  1443  19.21ht  1451  hbnt  1521  19.12  1533  19.38  1544  ax9o  1566  hbae  1584  equveli  1620  sb2  1628  drex1  1657  ax11b  1685  a16gb  1723  sb56  1743  sb6  1744  sbalyz  1853  hbsb4t  1867  moim  1942  mopick  1956
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