Axiom ax-12 2174

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 2082). 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 38870 and was replaced with this shorter ax-12 2174 in Jan. 2007. The old axiom is proved from this one as Theorem axc15 2424. Conversely, this axiom is proved from ax-c15 38870 as Theorem ax12 2425.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 38870) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 38870.

See ax12v 2175 and ax12v2 2176 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2130) 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 1959	. 2 wff 𝑥 = 𝑦
4		wph	. . . 4 wff 𝜑
5	4, 2	wal 1534	. . 3 wff ∀𝑦𝜑
6	3, 4	wi 4	. . . 4 wff (𝑥 = 𝑦 → 𝜑)
7	6, 1	wal 1534	. . 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  2175  equs5aALT  2366  equs5eALT  2367  axc11r  2368