Axiom ax-4 1524

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 1463. Conditional forms of the converse are given by ax12 1526, ax-16 1828, and ax-17 1540.

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 1789.

(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 1362	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  sp  1525  ax12  1526  hbequid  1527  spi  1550  hbim  1559  19.3h  1567  19.21h  1571  19.21bi  1572  hbimd  1587  19.21ht  1595  hbnt  1667  19.12  1679  19.38  1690  ax9o  1712  hbae  1732  equveli  1773  sb2  1781  drex1  1812  ax11b  1840  a16gb  1879  sb56  1900  sb6  1901  sbalyz  2018  hbsb4t  2032  moim  2109  mopick  2123