Axiom ax-4 2135

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 φ). (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 ∀yx = y.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2135 through ax-7 1734 plus rule ax-gen 1546). 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 1546. Conditional forms of the converse are given by ax-12 1925, ax-15 2143, ax-16 2144, and ax-17 1616.

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 2024.

An interesting alternate axiomatization uses ax467 2169 and ax-5o 2136 in place of ax-4 2135, ax-5 1557, ax-6 1729, and ax-7 1734.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 1747. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-4	⊢ (∀xφ → φ)

Detailed syntax breakdown of Axiom ax-4

Step	Hyp	Ref	Expression
1		wph	. . 3 wff φ
2		vx	. . 3 setvar x
3	1, 2	wal 1540	. 2 wff ∀xφ
4	3, 1	wi 4	1 wff (∀xφ → φ)

This axiom is referenced by:  ax5  2146  ax6  2147  hba1-o  2149  hbae-o  2153  ax11  2155  ax12from12o  2156  equid1  2158  sps-o  2159  ax46  2162  ax67to6  2167  ax467  2169  ax11indalem  2197  ax11inda2ALT  2198