Axiom ax-c5 39382

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 1802. Conditional forms of the converse are given by ax-13 2380, ax-c14 39390, ax-c16 39391, and ax-5 1917.

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 2079).

An interesting alternate axiomatization uses axc5c711 39417 and ax-c4 39383 in place of ax-c5 39382, ax-4 1816, ax-10 2152, and ax-11 2168.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2195. (Contributed by NM, 3-Jan-1993.) Use sp 2195 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 1545	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  39393  ax10fromc7  39394  hba1-o  39396  equid1  39398  hbae-o  39402  ax12fromc15  39404  ax13fromc9  39405  sps-o  39407  axc5c7  39410  axc711toc7  39415  axc5c711  39417  ax12indalem  39444  ax12inda2ALT  39445