Axiom ax-c5 39139

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 1796. Conditional forms of the converse are given by ax-13 2376, ax-c14 39147, ax-c16 39148, and ax-5 1911.

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 2073).

An interesting alternate axiomatization uses axc5c711 39174 and ax-c4 39140 in place of ax-c5 39139, ax-4 1810, ax-10 2146, and ax-11 2162.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2190. (Contributed by NM, 3-Jan-1993.) Use sp 2190 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  39150  ax10fromc7  39151  hba1-o  39153  equid1  39155  hbae-o  39159  ax12fromc15  39161  ax13fromc9  39162  sps-o  39164  axc5c7  39167  axc711toc7  39172  axc5c711  39174  ax12indalem  39201  ax12inda2ALT  39202