Axiom ax-4 1498

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 1437. Conditional forms of the converse are given by ax12 1500, ax-16 1802, and ax-17 1514.

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 1763.

(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 1341	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  sp  1499  ax12  1500  hbequid  1501  spi  1524  hbim  1533  19.3h  1541  19.21h  1545  19.21bi  1546  hbimd  1561  19.21ht  1569  hbnt  1641  19.12  1653  19.38  1664  ax9o  1686  hbae  1706  equveli  1747  sb2  1755  drex1  1786  ax11b  1814  a16gb  1853  sb56  1873  sb6  1874  sbalyz  1987  hbsb4t  2001  moim  2078  mopick  2092