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Theorem euind 3023
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
Hypotheses
Ref Expression
euind.1
euind.2
euind.3
Assertion
Ref Expression
euind
Distinct variable groups:   ,,   ,,   ,,   ,,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem euind
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 euind.2 . . . . . 6
21cbvexv 2003 . . . . 5
3 euind.1 . . . . . . . . 9
43isseti 2865 . . . . . . . 8
54biantrur 492 . . . . . . 7
65exbii 1582 . . . . . 6
7 19.41v 1901 . . . . . . . 8
87exbii 1582 . . . . . . 7
9 excom 1741 . . . . . . 7
108, 9bitr3i 242 . . . . . 6
116, 10bitri 240 . . . . 5
122, 11bitri 240 . . . 4
13 eqeq2 2362 . . . . . . . . 9
1413imim2i 13 . . . . . . . 8
15 bi2 189 . . . . . . . . . 10
1615imim2i 13 . . . . . . . . 9
17 an31 775 . . . . . . . . . . 11
1817imbi1i 315 . . . . . . . . . 10
19 impexp 433 . . . . . . . . . 10
20 impexp 433 . . . . . . . . . 10
2118, 19, 203bitr3i 266 . . . . . . . . 9
2216, 21sylib 188 . . . . . . . 8
2314, 22syl 15 . . . . . . 7
24232alimi 1560 . . . . . 6
25 19.23v 1891 . . . . . . . 8
2625albii 1566 . . . . . . 7
27 19.21v 1890 . . . . . . 7
2826, 27bitri 240 . . . . . 6
2924, 28sylib 188 . . . . 5
3029eximdv 1622 . . . 4
3112, 30syl5bi 208 . . 3
3231imp 418 . 2
33 pm4.24 624 . . . . . . . 8
3433biimpi 186 . . . . . . 7
35 prth 554 . . . . . . 7
36 eqtr3 2372 . . . . . . 7
3734, 35, 36syl56 30 . . . . . 6
3837alanimi 1562 . . . . 5
39 19.23v 1891 . . . . . . 7
4039biimpi 186 . . . . . 6
4140com12 27 . . . . 5
4238, 41syl5 28 . . . 4
4342alrimivv 1632 . . 3
4443adantl 452 . 2
45 eqeq1 2359 . . . . 5
4645imbi2d 307 . . . 4
4746albidv 1625 . . 3
4847eu4 2243 . 2
4932, 44, 48sylanbrc 645 1
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  weu 2204  cvv 2859
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-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861
This theorem is referenced by: (None)
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