Axiom ax-c5 32418

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 1666. Conditional forms of the converse are given by ax-13 2054, ax-c14 32426, ax-c16 32427, and ax-5 1749.

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 2148.

An interesting alternate axiomatization uses axc5c711 32452 and ax-c4 32419 in place of ax-c5 32418, ax-4 1679, ax-10 1888, and ax-11 1893.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1911. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c5	|- ( A. x ph -> ph )

Detailed syntax breakdown of Axiom ax-c5

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3	1, 2	wal 1436	. 2 wff A. x ph
4	3, 1	wi 4	1 wff ( A. x ph -> ph )

This axiom is referenced by:  ax4  32429  ax10  32430  hba1-o  32432  hbae-o  32436  ax12  32438  ax13fromc9  32439  equid1  32441  sps-o  32442  axc5c7  32445  axc711toc7  32450  axc5c711  32452  ax12indalem  32479  ax12inda2ALT  32480