Axiom ax-11o 1816

Description: Axiom ax-11o 1816 ("o" for "old") was the original version of ax-11 1499, before it was discovered (in Jan. 2007) that the shorter ax-11 1499 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."

This axiom is redundant, as shown by Theorem ax11o 1815.

This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1815. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-11o	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Detailed syntax breakdown of Axiom ax-11o

Step	Hyp	Ref	Expression
1		vx	. . . . 5 setvar 𝑥
2		vy	. . . . 5 setvar 𝑦
3	1, 2	weq 1496	. . . 4 wff 𝑥 = 𝑦
4	3, 1	wal 1346	. . 3 wff ∀𝑥 𝑥 = 𝑦
5	4	wn 3	. 2 wff ¬ ∀𝑥 𝑥 = 𝑦
6		wph	. . . 4 wff 𝜑
7	3, 6	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
8	7, 1	wal 1346	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
9	6, 8	wi 4	. . 3 wff (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
10	3, 9	wi 4	. 2 wff (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
11	5, 10	wi 4	1 wff (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

This axiom is referenced by: (None)