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Axiom ax-4 1400
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 1338. Conditional forms of the converse are given by ax-12 1402, ax-16 1695, and ax-17 1419.

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 1658.

(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 1241 . 2 wff 𝑥𝜑
43, 1wi 4 1 wff (∀𝑥𝜑𝜑)
Colors of variables: wff set class
This axiom is referenced by:  sp  1401  ax-12  1402  hbequid  1406  spi  1429  hbim  1437  19.3h  1445  19.21h  1449  19.21bi  1450  hbimd  1465  19.21ht  1473  hbnt  1543  19.12  1555  19.38  1566  ax9o  1588  hbae  1606  equveli  1642  sb2  1650  drex1  1679  ax11b  1707  a16gb  1745  sb56  1765  sb6  1766  sbalyz  1875  hbsb4t  1889  moim  1964  mopick  1978
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