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Axiom ax-c5 31871
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 1637. Conditional forms of the converse are given by ax-13 2024, ax-c14 31879, ax-c16 31880, and ax-5 1723.

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 2116.

An interesting alternate axiomatization uses axc5c711 31905 and ax-c4 31872 in place of ax-c5 31871, ax-4 1650, ax-10 1859, and ax-11 1864.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1881. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

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

Detailed syntax breakdown of Axiom ax-c5
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  setvar  x
31, 2wal 1401 . 2  wff  A. x ph
43, 1wi 4 1  wff  ( A. x ph  ->  ph )
Colors of variables: wff setvar class
This axiom is referenced by:  ax4  31882  ax10  31883  hba1-o  31885  hbae-o  31889  ax12  31891  ax13fromc9  31892  equid1  31894  sps-o  31895  axc5c7  31898  axc711toc7  31903  axc5c711  31905  ax12indalem  31932  ax12inda2ALT  31933
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