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Theorem aev-o 2182
 Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2144. Version of aev 1991 using ax-10o 2139. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aev-o (x x = yz w = v)
Distinct variable group:   x,y

Proof of Theorem aev-o
Dummy variables u t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae-o 2153 . 2 (x x = yzx x = y)
2 hbae-o 2153 . . . 4 (x x = ytx x = y)
3 ax-8 1675 . . . . 5 (x = t → (x = yt = y))
43spimv 1990 . . . 4 (x x = yt = y)
52, 4alrimih 1565 . . 3 (x x = yt t = y)
6 ax-8 1675 . . . . . . . 8 (y = u → (y = tu = t))
7 equcomi 1679 . . . . . . . 8 (u = tt = u)
86, 7syl6 29 . . . . . . 7 (y = u → (y = tt = u))
98spimv 1990 . . . . . 6 (y y = tt = u)
109aecoms-o 2152 . . . . 5 (t t = yt = u)
1110a5i-o 2150 . . . 4 (t t = yt t = u)
12 hbae-o 2153 . . . . 5 (t t = uvt t = u)
13 ax-8 1675 . . . . . 6 (t = v → (t = uv = u))
1413spimv 1990 . . . . 5 (t t = uv = u)
1512, 14alrimih 1565 . . . 4 (t t = uv v = u)
16 aecom-o 2151 . . . 4 (v v = uu u = v)
1711, 15, 163syl 18 . . 3 (t t = yu u = v)
18 ax-8 1675 . . . 4 (u = w → (u = vw = v))
1918spimv 1990 . . 3 (u u = vw = v)
205, 17, 193syl 18 . 2 (x x = yw = v)
211, 20alrimih 1565 1 (x x = yz w = v)
 Colors of variables: wff setvar class Syntax hints:   → 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142 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:  a16g-o  2186
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