Axiom ax-4 1520

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 1459. Conditional forms of the converse are given by ax12 1522, ax-16 1824, and ax-17 1536.

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 1785.

(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 1361	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  sp  1521  ax12  1522  hbequid  1523  spi  1546  hbim  1555  19.3h  1563  19.21h  1567  19.21bi  1568  hbimd  1583  19.21ht  1591  hbnt  1663  19.12  1675  19.38  1686  ax9o  1708  hbae  1728  equveli  1769  sb2  1777  drex1  1808  ax11b  1836  a16gb  1875  sb56  1895  sb6  1896  sbalyz  2009  hbsb4t  2023  moim  2100  mopick  2114