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Theorem ax11 2155
 Description: Rederivation of axiom ax-11 1746 from ax-11o 2141, ax-10o 2139, and other older axioms. The proof does not require ax-16 2144 or ax-17 1616. See theorem ax11o 1994 for the derivation of ax-11o 2141 from ax-11 1746. An open problem is whether we can prove this using ax-10 2140 instead of ax-10o 2139. This proof uses newer axioms ax-5 1557 and ax-9 1654, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2136 and ax-9o 2138. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax11 (x = y → (yφx(x = yφ)))

Proof of Theorem ax11
StepHypRef Expression
1 biidd 228 . . . . 5 (x x = y → (φφ))
21dral1-o 2154 . . . 4 (x x = y → (xφyφ))
3 ax-1 5 . . . . 5 (φ → (x = yφ))
43alimi 1559 . . . 4 (xφx(x = yφ))
52, 4syl6bir 220 . . 3 (x x = y → (yφx(x = yφ)))
65a1d 22 . 2 (x x = y → (x = y → (yφx(x = yφ))))
7 ax-4 2135 . . 3 (yφφ)
8 ax-11o 2141 . . 3 x x = y → (x = y → (φx(x = yφ))))
97, 8syl7 63 . 2 x x = y → (x = y → (yφx(x = yφ))))
106, 9pm2.61i 156 1 (x = y → (yφx(x = yφ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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-11o 2141  ax-12o 2142 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  ax10o-o  2203
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