Axiom ax-c5 37748

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 1797. Conditional forms of the converse are given by ax-13 2371, ax-c14 37756, ax-c16 37757, 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 37783 and ax-c4 37749 in place of ax-c5 37748, ax-4 1811, 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This axiom is obsolete and should no longer be used.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 1539	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  37759  ax10fromc7  37760  hba1-o  37762  equid1  37764  hbae-o  37768  ax12fromc15  37770  ax13fromc9  37771  sps-o  37773  axc5c7  37776  axc711toc7  37781  axc5c711  37783  ax12indalem  37810  ax12inda2ALT  37811