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Axiom ax-4 1397
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 1335. Conditional forms of the converse are given by ax-12 1399, ax-16 1692, and ax-17 1416.

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 1655.

(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 setvar x
31, 2wal 1240 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1398  ax-12  1399  hbequid  1403  spi  1426  hbim  1434  19.3h  1442  19.21h  1446  19.21bi  1447  hbimd  1462  19.21ht  1470  hbnt  1540  19.12  1552  19.38  1563  ax9o  1585  hbae  1603  equveli  1639  sb2  1647  drex1  1676  ax11b  1704  a16gb  1742  sb56  1762  sb6  1763  sbalyz  1872  hbsb4t  1886  moim  1961  mopick  1975
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