Axiom ax-4 1559

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 1498. Conditional forms of the converse are given by ax12 1561, ax-16 1863, and ax-17 1575.

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 1824.

(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 1396	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  sp  1560  ax12  1561  hbequid  1562  spi  1585  hbim  1594  19.3h  1602  19.21h  1606  19.21bi  1607  hbimd  1622  19.21ht  1630  hbnt  1701  19.12  1713  19.38  1724  ax9o  1746  hbae  1766  equveli  1808  sb2  1816  drex1  1847  ax11b  1875  a16gb  1914  sb56  1935  sb6  1936  sbalyz  2053  hbsb4t  2067  moim  2145  mopick  2159