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Axiom ax-4 1441
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 1379. Conditional forms of the converse are given by ax-12 1443, ax-16 1737, and ax-17 1460.

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 1700.

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

Assertion
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 1283 . 2 wff 𝑥𝜑
43, 1wi 4 1 wff (∀𝑥𝜑𝜑)
Colors of variables: wff set class
This axiom is referenced by:  sp  1442  ax-12  1443  hbequid  1447  spi  1470  hbim  1478  19.3h  1486  19.21h  1490  19.21bi  1491  hbimd  1506  19.21ht  1514  hbnt  1584  19.12  1596  19.38  1607  ax9o  1629  hbae  1648  equveli  1684  sb2  1692  drex1  1721  ax11b  1749  a16gb  1788  sb56  1808  sb6  1809  sbalyz  1918  hbsb4t  1932  moim  2007  mopick  2021
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