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Theorem eqeu 2705
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1
Assertion
Ref Expression
eqeu
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem eqeu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5
21spcegv 2635 . . . 4
32imp 115 . . 3
433adant3 923 . 2
5 eqeq2 2046 . . . . . . 7
65imbi2d 219 . . . . . 6
76albidv 1702 . . . . 5
87spcegv 2635 . . . 4
98imp 115 . . 3
1093adant2 922 . 2
11 nfv 1418 . . 3  F/
1211eu3 1943 . 2
134, 10, 12sylanbrc 394 1
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   w3a 884  wal 1240   wceq 1242  wex 1378   wcel 1390  weu 1897
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by: (None)
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