Axiom ax-4 1490

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 1429. Conditional forms of the converse are given by ax12 1492, ax-16 1794, and ax-17 1506.

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 1755.

(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 1333	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  sp  1491  ax12  1492  hbequid  1493  spi  1516  hbim  1525  19.3h  1533  19.21h  1537  19.21bi  1538  hbimd  1553  19.21ht  1561  hbnt  1633  19.12  1645  19.38  1656  ax9o  1678  hbae  1698  equveli  1739  sb2  1747  drex1  1778  ax11b  1806  a16gb  1845  sb56  1865  sb6  1866  sbalyz  1979  hbsb4t  1993  moim  2070  mopick  2084