Theorem ax10o 1708

Description: Show that ax-10o 1709 can be derived from ax-10 1498. An open problem is whether this theorem can be derived from ax-10 1498 and the others when ax-11 1499 is replaced with ax-11o 1816. See Theorem ax10 1710 for the rederivation of ax-10 1498 from 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, 16-May-2008.)

Assertion

Ref	Expression
ax10o	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem ax10o

Step	Hyp	Ref	Expression
1		ax-10 1498	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		ax-11 1499	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3	2	equcoms 1701	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
4	3	sps 1530	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
5		pm2.27 40	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
6	5	al2imi 1451	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
7	1, 4, 6	sylsyld 58	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  hbae  1711  dral1  1723