Axiom ax-10o 1709

Description: Axiom ax-10o 1709 ("o" for "old") was the original version of ax-10 1498, before it was discovered (in May 2008) that the shorter ax-10 1498 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by Theorem ax10o 1708.

Normally, ax10o 1708 should be used rather than ax-10o 1709, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-10o	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Detailed syntax breakdown of Axiom ax-10o

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

This axiom is referenced by:  ax10  1710