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Theorem ax10o 1678
Description: Show that ax-10o 1679 can be derived from ax-10 1468. An open problem is whether this theorem can be derived from ax-10 1468 and the others when ax-11 1469 is replaced with ax-11o 1779. See theorem ax10 1680 for the rederivation of ax-10 1468 from ax10o 1678.

Normally, ax10o 1678 should be used rather than ax-10o 1679, 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 1468 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 ax-11 1469 . . . 4 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
32equcoms 1669 . . 3 (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
43sps 1502 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
5 pm2.27 40 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
65al2imi 1419 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
71, 4, 6sylsyld 58 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1408  ax-gen 1410  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  hbae  1681  dral1  1693
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