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Theorem ax10lem4 1941
 Description: Lemma for ax10 1944. Change bound variable. (Contributed by NM, 8-Jul-2016.)
Assertion
Ref Expression
ax10lem4 (x x = wy y = x)
Distinct variable groups:   x,w   y,w

Proof of Theorem ax10lem4
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ax10lem1 1936 . . . . . 6 (x x = wy y = w)
2 equequ1 1684 . . . . . . . 8 (z = x → (z = wx = w))
32dvelimv 1939 . . . . . . 7 y y = x → (x = wy x = w))
4 hba1 1786 . . . . . . . . 9 (y x = wyy x = w)
5 equequ2 1686 . . . . . . . . . 10 (x = w → (y = xy = w))
65sps 1754 . . . . . . . . 9 (y x = w → (y = xy = w))
74, 6albidh 1590 . . . . . . . 8 (y x = w → (y y = xy y = w))
87biimprd 214 . . . . . . 7 (y x = w → (y y = wy y = x))
93, 8syl6 29 . . . . . 6 y y = x → (x = w → (y y = wy y = x)))
101, 9syl7 63 . . . . 5 y y = x → (x = w → (x x = wy y = x)))
1110spsd 1755 . . . 4 y y = x → (x x = w → (x x = wy y = x)))
1211pm2.43d 44 . . 3 y y = x → (x x = wy y = x))
1312com12 27 . 2 (x x = w → (¬ y y = xy y = x))
1413pm2.18d 103 1 (x x = wy y = x)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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:  ax10lem5  1942
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