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Axiom ax-c15 35418
Description: Axiom ax-c15 35418 was the original version of ax-12 2104, before it was discovered (in Jan. 2007) that the shorter ax-12 2104 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). 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. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."

Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-c15 35418 (from which the ax-12 2104 instance follows by theorem ax12 2357.) The proof is by induction on formula length, using ax12eq 35470 and ax12el 35471 for the basis steps and ax12indn 35472, ax12indi 35473, and ax12inda 35477 for the induction steps. (This paragraph is true provided we use ax-c11 35416 in place of ax-c11n 35417.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2355, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Detailed syntax breakdown of Axiom ax-c15
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
2 vy . . . . 5 setvar 𝑦
31, 2weq 1922 . . . 4 wff 𝑥 = 𝑦
43, 1wal 1505 . . 3 wff 𝑥 𝑥 = 𝑦
54wn 3 . 2 wff ¬ ∀𝑥 𝑥 = 𝑦
6 wph . . . 4 wff 𝜑
73, 6wi 4 . . . . 5 wff (𝑥 = 𝑦𝜑)
87, 1wal 1505 . . . 4 wff 𝑥(𝑥 = 𝑦𝜑)
96, 8wi 4 . . 3 wff (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
103, 9wi 4 . 2 wff (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
115, 10wi 4 1 wff (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff setvar class
This axiom is referenced by:  ax12fromc15  35434
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