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Axiom ax-4 1333
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 φ). (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 yx = 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 1269. Conditional forms of the converse are given by ax-12 1335, ax-16 1581, and ax-17 1350.

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 1545.

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

Ref Expression
ax-4 (xφφ)

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 set x
31, 2wal 1266 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1334  ax-12  1335  hbequid  1339  a4i  1361  a4s  1362  a4sd  1363  hbim  1369  19.3  1377  19.21  1380  19.21bi  1383  hbimd  1395  19.21ht  1401  hbnt  1449  19.12  1460  19.38  1465  ax9o  1479  hbae  1496  equveli  1526  sb2  1537  drex1  1565  ax11b  1593  sbf3t  1614  a16gb  1628  sb56  1647  sb6  1648  sbalyz  1752  hbsb4t  1764  mopick  2132  2eu1  2150  dfsb2  2184
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