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Axiom ax-c5 34487
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.) 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 1762. Conditional forms of the converse are given by ax-13 2282, ax-c14 34495, ax-c16 34496, and ax-5 1879.

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 2381.

An interesting alternate axiomatization uses axc5c711 34522 and ax-c4 34488 in place of ax-c5 34487, ax-4 1777, ax-10 2059, and ax-11 2074.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2091. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c5 (∀𝑥𝜑𝜑)

Detailed syntax breakdown of Axiom ax-c5
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
31, 2wal 1521 . 2 wff 𝑥𝜑
43, 1wi 4 1 wff (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  ax4fromc4  34498  ax10fromc7  34499  hba1-o  34501  equid1  34503  hbae-o  34507  ax12fromc15  34509  ax13fromc9  34510  sps-o  34512  axc5c7  34515  axc711toc7  34520  axc5c711  34522  ax12indalem  34549  ax12inda2ALT  34550
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