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Axiom ax-4 2087
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.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 2087 through ax-7 1720 plus rule ax-gen 1536). 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 1536. Conditional forms of the converse are given by ax-12 1878, ax-15 2095, ax-16 2096, and ax-17 1606.

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 1977.

An interesting alternate axiomatization uses ax467 2121 and ax-5o 2088 in place of ax-4 2087, ax-5 1547, ax-6 1715, and ax-7 1720.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1728. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Ref Expression
ax-4  |-  ( A. x ph  ->  ph )

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
31, 2wal 1530 . 2  wff  A. x ph
43, 1wi 4 1  wff  ( A. x ph  ->  ph )
Colors of variables: wff set class
This axiom is referenced by:  ax5  2098  ax6  2099  hba1-o  2101  hbae-o  2105  ax11  2107  ax12from12o  2108  equid1  2110  sps-o  2111  ax46  2114  ax67to6  2119  ax467  2121  ax11indalem  2149  ax11inda2ALT  2150
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