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Axiom ax-12 1983
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 2312). 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 32694 and was replaced with this shorter ax-12 1983 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2187. Conversely, this axiom is proved from ax-c15 32694 as theorem ax12 2224.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 32694) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 1984 and ax12v2 1985 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1957) but is used as an auxiliary axiom scheme to achieve metalogical 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 1822 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1466 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1466 . . 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  1984  ax12vOLD  1986  ax12vOLDOLD  1987  axc11r  2073  equs5aALT  2120  equs5eALT  2121  axc15OLD  2227  bj-axc15v  31543
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