Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax-c5 Structured version   Visualization version   GIF version

Axiom ax-c5 35563
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 1778. Conditional forms of the converse are given by ax-13 2343, ax-c14 35571, ax-c16 35572, and ax-5 1889.

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 2045.

An interesting alternate axiomatization uses axc5c711 35598 and ax-c4 35564 in place of ax-c5 35563, ax-4 1792, ax-10 2111, and ax-11 2125.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2145. (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 1520 . 2 wff 𝑥𝜑
43, 1wi 4 1 wff (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  ax4fromc4  35574  ax10fromc7  35575  hba1-o  35577  equid1  35579  hbae-o  35583  ax12fromc15  35585  ax13fromc9  35586  sps-o  35588  axc5c7  35591  axc711toc7  35596  axc5c711  35598  ax12indalem  35625  ax12inda2ALT  35626
  Copyright terms: Public domain W3C validator