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Axiom ax-4 1338
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 1274. Conditional forms of the converse are given by ax-12 1340, ax-16 1613, and ax-17 1356.

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 1577.

(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 1271 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1339  ax-12  1340  hbequid  1344  spi  1367  hbim  1375  19.3h  1383  19.21h  1387  19.21bi  1390  hbimd  1402  19.21ht  1410  hbnt  1470  19.12  1481  19.38  1489  ax9o  1507  hbae  1524  equveli  1560  sb2  1569  drex1  1597  ax11b  1625  sbf3t  1646  a16gb  1662  sb56  1681  sb6  1682  sbalyz  1789  hbsb4t  1801  moim  1872  mopick  1887  2eu1  2664  dfsb2  2691
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