Axiom ax-4 1536

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 1475. Conditional forms of the converse are given by ax12 1538, ax-16 1840, and ax-17 1552.

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 1801.

(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 1373	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  sp  1537  ax12  1538  hbequid  1539  spi  1562  hbim  1571  19.3h  1579  19.21h  1583  19.21bi  1584  hbimd  1599  19.21ht  1607  hbnt  1679  19.12  1691  19.38  1702  ax9o  1724  hbae  1744  equveli  1785  sb2  1793  drex1  1824  ax11b  1852  a16gb  1891  sb56  1912  sb6  1913  sbalyz  2030  hbsb4t  2044  moim  2122  mopick  2136