Axiom ax-4 1556

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 1495. Conditional forms of the converse are given by ax12 1558, ax-16 1860, and ax-17 1572.

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 1821.

(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 1393	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  sp  1557  ax12  1558  hbequid  1559  spi  1582  hbim  1591  19.3h  1599  19.21h  1603  19.21bi  1604  hbimd  1619  19.21ht  1627  hbnt  1699  19.12  1711  19.38  1722  ax9o  1744  hbae  1764  equveli  1805  sb2  1813  drex1  1844  ax11b  1872  a16gb  1911  sb56  1932  sb6  1933  sbalyz  2050  hbsb4t  2064  moim  2142  mopick  2156