Axiom ax-4 1532

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 ph). (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 A. y x = 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 1471. Conditional forms of the converse are given by ax12 1534, ax-16 1836, and ax-17 1548.

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 1797.

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

Assertion

Ref	Expression
ax-4	|- ( A. x ph -> ph )

Detailed syntax breakdown of Axiom ax-4

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3	1, 2	wal 1370	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  sp  1533  ax12  1534  hbequid  1535  spi  1558  hbim  1567  19.3h  1575  19.21h  1579  19.21bi  1580  hbimd  1595  19.21ht  1603  hbnt  1675  19.12  1687  19.38  1698  ax9o  1720  hbae  1740  equveli  1781  sb2  1789  drex1  1820  ax11b  1848  a16gb  1887  sb56  1908  sb6  1909  sbalyz  2026  hbsb4t  2040  moim  2117  mopick  2131