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Theorem ax10o 1581
 Description: Show that ax-10o 1582 can be derived from ax-10 1373. An open problem is whether this theorem can be derived from ax-10 1373 and the others when ax-11 1374 is replaced with ax-11o 1682. See theorem ax10 1583 for the rederivation of ax-10 1373 from ax10o 1581. Normally, ax10o 1581 should be used rather than ax-10o 1582, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
Assertion
Ref Expression
ax10o (x x = y → (xφyφ))

Proof of Theorem ax10o
StepHypRef Expression
1 ax-10 1373 . 2 (x x = yy y = x)
2 ax-11 1374 . . . 4 (y = x → (xφy(y = xφ)))
32equcoms 1572 . . 3 (x = y → (xφy(y = xφ)))
43sps 1408 . 2 (x x = y → (xφy(y = xφ)))
5 pm2.27 35 . . 3 (y = x → ((y = xφ) → φ))
65al2imi 1323 . 2 (y y = x → (y(y = xφ) → yφ))
71, 4, 6sylsyld 52 1 (x x = y → (xφyφ))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-5 1312  ax-gen 1314  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  hbae  1584  dral1  1596
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