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Axiom ax-12 2169
Description: Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent 𝑥(𝑥 = 𝑦𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2086). 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 35892 and was replaced with this shorter ax-12 2169 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2440. Conversely, this axiom is proved from ax-c15 35892 as theorem ax12 2442.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 35892) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 35892.

See ax12v 2170 and ax12v2 2171 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 = 𝑦𝜑)))

Detailed syntax breakdown of Axiom ax-12
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1957 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1528 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1528 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
85, 7wi 4 . 2 wff (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
93, 8wi 4 1 wff (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
This axiom is referenced by:  ax12v  2170  equs5aALT  2379  equs5eALT  2380  axc11r  2381
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