Theorem ax10o 1715

Description: Show that ax-10o 1716 can be derived from ax-10 1505. An open problem is whether this theorem can be derived from ax-10 1505 and the others when ax-11 1506 is replaced with ax-11o 1823. See Theorem ax10 1717 for the rederivation of ax-10 1505 from ax10o 1715.

Normally, ax10o 1715 should be used rather than ax-10o 1716, 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 1505	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		ax-11 1506	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3	2	equcoms 1708	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
4	3	sps 1537	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
5		pm2.27 40	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
6	5	al2imi 1458	. 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 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbae  1718  dral1  1730