Axiom ax-4 1468

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 1406. Conditional forms of the converse are given by ax-12 1470, ax-16 1766, and ax-17 1487.

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 1729.

(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 1310	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  sp  1469  ax-12  1470  hbequid  1474  spi  1497  hbim  1505  19.3h  1513  19.21h  1517  19.21bi  1518  hbimd  1533  19.21ht  1541  hbnt  1612  19.12  1624  19.38  1635  ax9o  1657  hbae  1677  equveli  1713  sb2  1721  drex1  1750  ax11b  1778  a16gb  1817  sb56  1837  sb6  1838  sbalyz  1948  hbsb4t  1962  moim  2037  mopick  2051