Axiom ax-12 2177

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 2093). 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 36040 and was replaced with this shorter ax-12 2177 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2444. Conversely, this axiom is proved from ax-c15 36040 as theorem ax12 2445.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 36040) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 36040.

See ax12v 2178 and ax12v2 2179 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2137) 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 1964	. 2 wff 𝑥 = 𝑦
4		wph	. . . 4 wff 𝜑
5	4, 2	wal 1535	. . 3 wff ∀𝑦𝜑
6	3, 4	wi 4	. . . 4 wff (𝑥 = 𝑦 → 𝜑)
7	6, 1	wal 1535	. . 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  2178  equs5aALT  2384  equs5eALT  2385  axc11r  2386