Axiom ax-4 1534

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 1473. Conditional forms of the converse are given by ax12 1536, ax-16 1838, and ax-17 1550.

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 1799.

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

Assertion

Ref	Expression
ax-4	⊢ (∀𝑥𝜑 → 𝜑)

Detailed syntax breakdown of Axiom ax-4

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wal 1371	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  sp  1535  ax12  1536  hbequid  1537  spi  1560  hbim  1569  19.3h  1577  19.21h  1581  19.21bi  1582  hbimd  1597  19.21ht  1605  hbnt  1677  19.12  1689  19.38  1700  ax9o  1722  hbae  1742  equveli  1783  sb2  1791  drex1  1822  ax11b  1850  a16gb  1889  sb56  1910  sb6  1911  sbalyz  2028  hbsb4t  2042  moim  2119  mopick  2133