Axiom ax-c5 39343

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 2377, ax-c14 39351, ax-c16 39352, and ax-5 1912.

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 2074).

An interesting alternate axiomatization uses axc5c711 39378 and ax-c4 39344 in place of ax-c5 39343, ax-4 1811, ax-10 2147, and ax-11 2163.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2191. (Contributed by NM, 3-Jan-1993.) Use sp 2191 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 1540	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  39354  ax10fromc7  39355  hba1-o  39357  equid1  39359  hbae-o  39363  ax12fromc15  39365  ax13fromc9  39366  sps-o  39368  axc5c7  39371  axc711toc7  39376  axc5c711  39378  ax12indalem  39405  ax12inda2ALT  39406