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Axiom ax-c5 2214
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 , it is true for any specific (that would typically occur as a free variable in the wff substituted for ). (A free variable is one that does not occur in the scope of a quantifier: and are both free in , but only is free in .) 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 1618. Conditional forms of the converse are given by ax-13 1999, ax-c14 2222, ax-c16 2223, and ax-5 1704.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 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 2094.

An interesting alternate axiomatization uses axc5c711 2248 and ax-c4 2215 in place of ax-c5 2214, ax-4 1631, ax-10 1837, and ax-11 1842.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1859. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c5

Detailed syntax breakdown of Axiom ax-c5
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2wal 1393 . 2
43, 1wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  ax4  2225  ax10  2226  hba1-o  2228  hbae-o  2232  ax12  2234  ax13fromc9  2235  equid1  2237  sps-o  2238  axc5c7  2241  axc711toc7  2246  axc5c711  2248  ax12indalem  2275  ax12inda2ALT  2276
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