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Axiom ax-c15 33651
Description: Axiom ax-c15 33651 was the original version of ax-12 2044, before it was discovered (in Jan. 2007) that the shorter ax-12 2044 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 33651 (from which the ax-12 2044 instance follows by theorem ax12 2303.) The proof is by induction on formula length, using ax12eq 33703 and ax12el 33704 for the basis steps and ax12indn 33705, ax12indi 33706, and ax12inda 33710 for the induction steps. (This paragraph is true provided we use ax-c11 33649 in place of ax-c11n 33650.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2302, 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 1871 . . . 4 wff 𝑥 = 𝑦
43, 1wal 1478 . . 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 1478 . . . 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  33667
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