Axiom ax-c5 39697

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 1828. Conditional forms of the converse are given by ax-13 2407, ax-c14 39705, ax-c16 39706, and ax-5 1943.

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 2105).

An interesting alternate axiomatization uses axc5c711 39732 and ax-c4 39698 in place of ax-c5 39697, ax-4 1842, ax-10 2179, and ax-11 2195.

This axiom is obsolete and should no longer be used. It is proved above as Theorem sp 2222. (Contributed by NM, 3-Jan-1993.) Use sp 2222 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 1568	. 2 wff ∀𝑥𝜑
4	3, 1	wi 4	1 wff (∀𝑥𝜑 → 𝜑)

This axiom is used by:  ax4fromc4  39708  ax10fromc7  39709  hba1-o  39711  equid1  39713  hbae-o  39717  ax12fromc15  39719  ax13fromc9  39720  sps-o  39722  axc5c7  39725  axc711toc7  39730  axc5c711  39732  ax12indalem  39759  ax12inda2ALT  39760