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Axiom ax-12 2034
 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 2417). 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 33192 and was replaced with this shorter ax-12 2034 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2291. Conversely, this axiom is proved from ax-c15 33192 as theorem ax12 2292. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 33192) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax12v 2035 and ax12v2 2036 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 1997) 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 1861 . 2 wff 𝑥 = 𝑦
4 wph . . . 4 wff 𝜑
54, 2wal 1473 . . 3 wff 𝑦𝜑
63, 4wi 4 . . . 4 wff (𝑥 = 𝑦𝜑)
76, 1wal 1473 . . 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  2035  ax12vOLD  2037  ax12vOLDOLD  2038  equs5aALT  2165  equs5eALT  2166  axc11r  2175  axc15OLD  2332
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