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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
StepHypRef Expression
1 ax-10 1396 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 ax-11 1397 . . . 4 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
32equcoms 1594 . . 3 (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
43sps 1430 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
5 pm2.27 35 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
65al2imi 1347 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
71, 4, 6sylsyld 52 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 Colors of variables: wff set class 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
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