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Theorem hbae-o 2153
 Description: All variables are effectively bound in an identical variable specifier. Version of hbae 1953 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbae-o (x x = yzx x = y)

Proof of Theorem hbae-o
StepHypRef Expression
1 ax-4 2135 . . . . 5 (x x = yx = y)
2 ax-12o 2142 . . . . 5 z z = x → (¬ z z = y → (x = yz x = y)))
31, 2syl7 63 . . . 4 z z = x → (¬ z z = y → (x x = yz x = y)))
4 ax-10o 2139 . . . . 5 (x x = z → (x x = yz x = y))
54aecoms-o 2152 . . . 4 (z z = x → (x x = yz x = y))
6 ax-10o 2139 . . . . . . 7 (x x = y → (x x = yy x = y))
76pm2.43i 43 . . . . . 6 (x x = yy x = y)
8 ax-10o 2139 . . . . . 6 (y y = z → (y x = yz x = y))
97, 8syl5 28 . . . . 5 (y y = z → (x x = yz x = y))
109aecoms-o 2152 . . . 4 (z z = y → (x x = yz x = y))
113, 5, 10pm2.61ii 157 . . 3 (x x = yz x = y)
1211a5i-o 2150 . 2 (x x = yxz x = y)
13 ax-7 1734 . 2 (xz x = yzx x = y)
1412, 13syl 15 1 (x x = yzx x = y)
 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-12o 2142 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  dral1-o  2154  hbnae-o  2179  dral2-o  2181  aev-o  2182
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