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Theorem ax10o-o 2203
 Description: Show that ax-10o 2139 can be derived from ax-10 2140. An open problem is whether this theorem can be derived from ax-10 2140 and the others when ax-11 1746 is replaced with ax-11o 2141. See theorem ax10from10o 2177 for the rederivation of ax-10 2140 from ax10o 1952. Normally, ax10o 1952 should be used rather than ax-10o 2139 or ax10o-o 2203, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax10o-o (x x = y → (xφyφ))

Proof of Theorem ax10o-o
StepHypRef Expression
1 ax-10 2140 . 2 (x x = yy y = x)
2 ax11 2155 . . . 4 (y = x → (xφy(y = xφ)))
32equcoms 1681 . . 3 (x = y → (xφy(y = xφ)))
43sps-o 2159 . 2 (x x = y → (xφy(y = xφ)))
5 pm2.27 35 . . 3 (y = x → ((y = xφ) → φ))
65al2imi 1561 . 2 (y y = x → (y(y = xφ) → yφ))
71, 4, 6sylsyld 52 1 (x x = y → (xφyφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-10 2140  ax-11o 2141  ax-12o 2142 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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