Axiom ax-c5 38256

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 1789. Conditional forms of the converse are given by ax-13 2363, ax-c14 38264, ax-c16 38265, and ax-5 1905.

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 2063).

An interesting alternate axiomatization uses axc5c711 38291 and ax-c4 38257 in place of ax-c5 38256, ax-4 1803, ax-10 2129, and ax-11 2146.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2168. (Contributed by NM, 3-Jan-1993.) Use sp 2168 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 1531	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  38267  ax10fromc7  38268  hba1-o  38270  equid1  38272  hbae-o  38276  ax12fromc15  38278  ax13fromc9  38279  sps-o  38281  axc5c7  38284  axc711toc7  38289  axc5c711  38291  ax12indalem  38318  ax12inda2ALT  38319