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Theorem hbae 1953
 Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbae (x x = yzx x = y)

Proof of Theorem hbae
StepHypRef Expression
1 sp 1747 . . . . 5 (x x = yx = y)
2 ax12o 1934 . . . . 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 ax10o 1952 . . . . 5 (x x = z → (x x = yz x = y))
54aecoms 1947 . . . 4 (z z = x → (x x = yz x = y))
6 ax10o 1952 . . . . . . 7 (x x = y → (x x = yy x = y))
76pm2.43i 43 . . . . . 6 (x x = yy x = y)
8 ax10o 1952 . . . . . 6 (y y = z → (y x = yz x = y))
97, 8syl5 28 . . . . 5 (y y = z → (x x = yz x = y))
109aecoms 1947 . . . 4 (z z = y → (x x = yz x = y))
113, 5, 10pm2.61ii 157 . . 3 (x x = yz x = y)
1211a5i 1789 . 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-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfae  1954  hbnae  1955  dral1  1965  dral2  1966  drex2  1968  aev  1991
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