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Axiom ax-c5 36198
 Description: Axiom of Specialization. A universally quantified wff implies the wff without the universal quantifier (i.e., an instance, or special case, of the generalized wff). In other words, if something is true for all 𝑥, then 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 1797. Conditional forms of the converse are given by ax-13 2379, ax-c14 36206, ax-c16 36207, and ax-5 1911. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. In our axiomatization, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution (see stdpc4 2073). An interesting alternate axiomatization uses axc5c711 36233 and ax-c4 36199 in place of ax-c5 36198, ax-4 1811, ax-10 2142, and ax-11 2158. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2180. (Contributed by NM, 3-Jan-1993.) Use sp 2180 instead. (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 1536 . 2 wff 𝑥𝜑
43, 1wi 4 1 wff (∀𝑥𝜑𝜑)
 Colors of variables: wff setvar class This axiom is referenced by:  ax4fromc4  36209  ax10fromc7  36210  hba1-o  36212  equid1  36214  hbae-o  36218  ax12fromc15  36220  ax13fromc9  36221  sps-o  36223  axc5c7  36226  axc711toc7  36231  axc5c711  36233  ax12indalem  36260  ax12inda2ALT  36261
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