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Theorem euind 2952
 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 1943 . . . . 5
3 euind.1 . . . . . . . . 9
43isseti 2794 . . . . . . . 8
54biantrur 492 . . . . . . 7
65exbii 1569 . . . . . 6
7 19.41v 1842 . . . . . . . 8
87exbii 1569 . . . . . . 7
9 excom 1786 . . . . . . 7
108, 9bitr3i 242 . . . . . 6
116, 10bitri 240 . . . . 5
122, 11bitri 240 . . . 4
13 eqeq2 2292 . . . . . . . . 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 1547 . . . . . 6
25 19.23v 1832 . . . . . . . 8
2625albii 1553 . . . . . . 7
27 19.21v 1831 . . . . . . 7
2826, 27bitri 240 . . . . . 6
2924, 28sylib 188 . . . . 5
3029eximdv 1608 . . . 4
3112, 30syl5bi 208 . . 3
3231imp 418 . 2
33 pm4.24 624 . . . . . . . 8
3433biimpi 186 . . . . . . 7
35 prth 554 . . . . . . 7
36 eqtr3 2302 . . . . . . 7
3734, 35, 36syl56 30 . . . . . 6
3837alanimi 1549 . . . . 5
39 19.23v 1832 . . . . . . 7
4039biimpi 186 . . . . . 6
4140com12 27 . . . . 5
4238, 41syl5 28 . . . 4
4342alrimivv 1618 . . 3
4443adantl 452 . 2
45 eqeq1 2289 . . . . 5
4645imbi2d 307 . . . 4
4746albidv 1611 . . 3
4847eu4 2182 . 2
4932, 44, 48sylanbrc 645 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1527  wex 1528   wceq 1623   wcel 1684  weu 2143  cvv 2788 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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