Axiom ax-c5 36897

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 1798. Conditional forms of the converse are given by ax-13 2372, ax-c14 36905, ax-c16 36906, and ax-5 1913.

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 2071).

An interesting alternate axiomatization uses axc5c711 36932 and ax-c4 36898 in place of ax-c5 36897, ax-4 1812, ax-10 2137, and ax-11 2154.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2176. (Contributed by NM, 3-Jan-1993.) Use sp 2176 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 1537	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  36908  ax10fromc7  36909  hba1-o  36911  equid1  36913  hbae-o  36917  ax12fromc15  36919  ax13fromc9  36920  sps-o  36922  axc5c7  36925  axc711toc7  36930  axc5c711  36932  ax12indalem  36959  ax12inda2ALT  36960