Axiom ax-12 2164

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 2081). 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 38285 and was replaced with this shorter ax-12 2164 in Jan. 2007. The old axiom is proved from this one as Theorem axc15 2416. Conversely, this axiom is proved from ax-c15 38285 as Theorem ax12 2417.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 38285) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html 38285.

See ax12v 2165 and ax12v2 2166 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2122) 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 1532	. . 3 wff ∀𝑦𝜑
6	3, 4	wi 4	. . . 4 wff (𝑥 = 𝑦 → 𝜑)
7	6, 1	wal 1532	. . 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  2165  equs5aALT  2358  equs5eALT  2359  axc11r  2360