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Theorem aev 1671
 Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1673. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
aev (x x = yz w = v)
Distinct variable group:   x,y

Proof of Theorem aev
Dummy variables u f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae 1584 . 2 (x x = yzx x = y)
2 hbae 1584 . . . 4 (x x = yfx x = y)
3 ax-8 1372 . . . . 5 (x = f → (x = yf = y))
43spimv 1670 . . . 4 (x x = yf = y)
52, 4alrimih 1334 . . 3 (x x = yf f = y)
6 ax-8 1372 . . . . . . . 8 (y = u → (y = fu = f))
7 equcomi 1570 . . . . . . . 8 (u = ff = u)
86, 7syl6 29 . . . . . . 7 (y = u → (y = ff = u))
98spimv 1670 . . . . . 6 (y y = ff = u)
109alequcoms 1386 . . . . 5 (f f = yf = u)
1110a5i 1413 . . . 4 (f f = yf f = u)
12 hbae 1584 . . . . 5 (f f = uvf f = u)
13 ax-8 1372 . . . . . 6 (f = v → (f = uv = u))
1413spimv 1670 . . . . 5 (f f = uv = u)
1512, 14alrimih 1334 . . . 4 (f f = uv v = u)
16 alequcom 1385 . . . 4 (v v = uu u = v)
1711, 15, 163syl 17 . . 3 (f f = yu u = v)
18 ax-8 1372 . . . 4 (u = w → (u = vw = v))
1918spimv 1670 . . 3 (u u = vw = v)
205, 17, 193syl 17 . 2 (x x = yw = v)
211, 20alrimih 1334 1 (x x = yz w = v)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326 This theorem is referenced by:  ax16  1672  a16g  1722
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