Axiom ax-c5 38349

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 1790. Conditional forms of the converse are given by ax-13 2367, ax-c14 38357, ax-c16 38358, and ax-5 1906.

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 2064).

An interesting alternate axiomatization uses axc5c711 38384 and ax-c4 38350 in place of ax-c5 38349, ax-4 1804, ax-10 2130, and ax-11 2147.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2172. (Contributed by NM, 3-Jan-1993.) Use sp 2172 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 1532	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax4fromc4  38360  ax10fromc7  38361  hba1-o  38363  equid1  38365  hbae-o  38369  ax12fromc15  38371  ax13fromc9  38372  sps-o  38374  axc5c7  38377  axc711toc7  38382  axc5c711  38384  ax12indalem  38411  ax12inda2ALT  38412