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Axiom ax-4 1373
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 1311. Conditional forms of the converse are given by ax-12 1375, ax-16 1668, and ax-17 1392.

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 1631.

(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 1221 . 2
43, 1wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  sp  1374  ax-12  1375  hbequid  1379  spi  1402  hbim  1410  19.3h  1418  19.21h  1422  19.21bi  1423  hbimd  1438  19.21ht  1446  hbnt  1516  19.12  1528  19.38  1539  ax9o  1561  hbae  1579  equveli  1615  sb2  1623  drex1  1652  ax11b  1680  a16gb  1718  sb56  1738  sb6  1739  sbalyz  1848  hbsb4t  1862  moim  1937  mopick  1951
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