Axiom ax-12 2180

Description: Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀𝑥(𝑥 = 𝑦 → 𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2088). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-c15 38927 and was replaced with this shorter ax-12 2180 in Jan. 2007. The old axiom is proved from this one as Theorem axc15 2422. Conversely, this axiom is proved from ax-c15 38927 as Theorem ax12 2423.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 38927) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 38927.

See ax12v 2181 and ax12v2 2182 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2136) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-12	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Detailed syntax breakdown of Axiom ax-12

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		vy	. . 3 setvar 𝑦
3	1, 2	weq 1963	. 2 wff 𝑥 = 𝑦
4		wph	. . . 4 wff 𝜑
5	4, 2	wal 1539	. . 3 wff ∀𝑦𝜑
6	3, 4	wi 4	. . . 4 wff (𝑥 = 𝑦 → 𝜑)
7	6, 1	wal 1539	. . 3 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
8	5, 7	wi 4	. 2 wff (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
9	3, 8	wi 4	1 wff (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

This axiom is referenced by:  ax12v  2181  equs5aALT  2366  equs5eALT  2367  axc11r  2368