Axiom ax-c5 38864

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 1791. Conditional forms of the converse are given by ax-13 2374, ax-c14 38872, ax-c16 38873, and ax-5 1907.

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 2065).

An interesting alternate axiomatization uses axc5c711 38899 and ax-c4 38865 in place of ax-c5 38864, ax-4 1805, ax-10 2138, and ax-11 2154.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2180. (Contributed by NM, 3-Jan-1993.) Use sp 2180 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 1534	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  38875  ax10fromc7  38876  hba1-o  38878  equid1  38880  hbae-o  38884  ax12fromc15  38886  ax13fromc9  38887  sps-o  38889  axc5c7  38892  axc711toc7  38897  axc5c711  38899  ax12indalem  38926  ax12inda2ALT  38927