Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  euind Unicode version

Theorem euind 3286
 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:   ,,   ,,   ,,   ,,   ,

Proof of Theorem euind
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 euind.2 . . . . . 6
21cbvexv 2024 . . . . 5
3 euind.1 . . . . . . . . 9
43isseti 3115 . . . . . . . 8
54biantrur 506 . . . . . . 7
65exbii 1667 . . . . . 6
7 19.41v 1771 . . . . . . 7
87exbii 1667 . . . . . 6
9 excom 1849 . . . . . 6
106, 8, 93bitr2i 273 . . . . 5
112, 10bitri 249 . . . 4
12 eqeq2 2472 . . . . . . . . 9
1312imim2i 14 . . . . . . . 8
14 bi2 198 . . . . . . . . . 10
1514imim2i 14 . . . . . . . . 9
16 an31 800 . . . . . . . . . . 11
1716imbi1i 325 . . . . . . . . . 10
18 impexp 446 . . . . . . . . . 10
19 impexp 446 . . . . . . . . . 10
2017, 18, 193bitr3i 275 . . . . . . . . 9
2115, 20sylib 196 . . . . . . . 8
2213, 21syl 16 . . . . . . 7
23222alimi 1634 . . . . . 6
24 19.23v 1760 . . . . . . . 8
2524albii 1640 . . . . . . 7
26 19.21v 1729 . . . . . . 7
2725, 26bitri 249 . . . . . 6
2823, 27sylib 196 . . . . 5
2928eximdv 1710 . . . 4
3011, 29syl5bi 217 . . 3
3130imp 429 . 2
32 pm4.24 643 . . . . . . . 8
3332biimpi 194 . . . . . . 7
34 prth 571 . . . . . . 7
35 eqtr3 2485 . . . . . . 7
3633, 34, 35syl56 34 . . . . . 6
3736alanimi 1637 . . . . 5
38 19.23v 1760 . . . . . . 7
3938biimpi 194 . . . . . 6
4039com12 31 . . . . 5
4137, 40syl5 32 . . . 4
4241alrimivv 1720 . . 3
4342adantl 466 . 2
44 eqeq1 2461 . . . . 5
4544imbi2d 316 . . . 4
4645albidv 1713 . . 3
4746eu4 2338 . 2
4831, 43, 47sylanbrc 664 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282   cvv 3109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
 Copyright terms: Public domain W3C validator