Axiom ax-c5 38839

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 1793. Conditional forms of the converse are given by ax-13 2380, ax-c14 38847, ax-c16 38848, and ax-5 1909.

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 2068).

An interesting alternate axiomatization uses axc5c711 38874 and ax-c4 38840 in place of ax-c5 38839, ax-4 1807, ax-10 2141, and ax-11 2158.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2184. (Contributed by NM, 3-Jan-1993.) Use sp 2184 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax-c5	⊢ (∀𝑥𝜑 → 𝜑)

Detailed syntax breakdown of Axiom ax-c5

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wal 1535	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  38850  ax10fromc7  38851  hba1-o  38853  equid1  38855  hbae-o  38859  ax12fromc15  38861  ax13fromc9  38862  sps-o  38864  axc5c7  38867  axc711toc7  38872  axc5c711  38874  ax12indalem  38901  ax12inda2ALT  38902