Theorem ax10o 1603

Description: Show that ax-10o 1604 can be derived from ax-10 1396. An open problem is whether this theorem can be derived from ax-10 1396 and the others when ax-11 1397 is replaced with ax-11o 1704. See theorem ax10 1605 for the rederivation of ax-10 1396 from ax10o 1603.

Normally, ax10o 1603 should be used rather than ax-10o 1604, 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 1396	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		ax-11 1397	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3	2	equcoms 1594	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
4	3	sps 1430	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
5		pm2.27 35	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
6	5	al2imi 1347	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
7	1, 4, 6	sylsyld 52	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  hbae  1606  dral1  1618