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Axiom ax-c15 34493
Description: Axiom ax-c15 34493 was the original version of ax-12 2087, before it was discovered (in Jan. 2007) that the shorter ax-12 2087 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 34493 (from which the ax-12 2087 instance follows by theorem ax12 2340.) The proof is by induction on formula length, using ax12eq 34545 and ax12el 34546 for the basis steps and ax12indn 34547, ax12indi 34548, and ax12inda 34552 for the induction steps. (This paragraph is true provided we use ax-c11 34491 in place of ax-c11n 34492.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2339, 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 1931 . . . 4 wff 𝑥 = 𝑦
43, 1wal 1521 . . 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 1521 . . . 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  34509
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