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Axiom ax-c15 36178
Description: Axiom ax-c15 36178 was the original version of ax-12 2176, before it was discovered (in Jan. 2007) that the shorter ax-12 2176 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 36178 (from which the ax-12 2176 instance follows by theorem ax12 2437.) The proof is by induction on formula length, using ax12eq 36230 and ax12el 36231 for the basis steps and ax12indn 36232, ax12indi 36233, and ax12inda 36237 for the induction steps. (This paragraph is true provided we use ax-c11 36176 in place of ax-c11n 36177.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2436, 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 1964 . . . 4 wff 𝑥 = 𝑦
43, 1wal 1536 . . 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 1536 . . . 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  36194
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