Axiom ax-c5 39519

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 1818. Conditional forms of the converse are given by ax-13 2406, ax-c14 39527, ax-c16 39528, and ax-5 1933.

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 2101).

An interesting alternate axiomatization uses axc5c711 39554 and ax-c4 39520 in place of ax-c5 39519, ax-4 1832, ax-10 2178, and ax-11 2194.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2221. (Contributed by NM, 3-Jan-1993.) Use sp 2221 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 1561	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  39530  ax10fromc7  39531  hba1-o  39533  equid1  39535  hbae-o  39539  ax12fromc15  39541  ax13fromc9  39542  sps-o  39544  axc5c7  39547  axc711toc7  39552  axc5c711  39554  ax12indalem  39581  ax12inda2ALT  39582