Axiom ax-c5 36023

Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all 𝑥, 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 1795. Conditional forms of the converse are given by ax-13 2389, ax-c14 36031, ax-c16 36032, and ax-5 1910.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2072.

An interesting alternate axiomatization uses axc5c711 36058 and ax-c4 36024 in place of ax-c5 36023, ax-4 1809, ax-10 2144, and ax-11 2160.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2181. (Contributed by NM, 3-Jan-1993.) (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  36034  ax10fromc7  36035  hba1-o  36037  equid1  36039  hbae-o  36043  ax12fromc15  36045  ax13fromc9  36046  sps-o  36048  axc5c7  36051  axc711toc7  36056  axc5c711  36058  ax12indalem  36085  ax12inda2ALT  36086