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Theorem preqsn 4213
 Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
preqsn.1
preqsn.2
preqsn.3
Assertion
Ref Expression
preqsn

Proof of Theorem preqsn
StepHypRef Expression
1 dfsn2 4042 . . 3
21eqeq2i 2475 . 2
3 preqsn.1 . . . 4
4 preqsn.2 . . . 4
5 preqsn.3 . . . 4
63, 4, 5, 5preq12b 4206 . . 3
7 oridm 514 . . . 4
8 eqtr3 2485 . . . . . 6
9 simpr 461 . . . . . 6
108, 9jca 532 . . . . 5
11 eqtr 2483 . . . . . 6
12 simpr 461 . . . . . 6
1311, 12jca 532 . . . . 5
1410, 13impbii 188 . . . 4
157, 14bitri 249 . . 3
166, 15bitri 249 . 2
172, 16bitri 249 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  {csn 4029  {cpr 4031 This theorem is referenced by:  opeqsn  4748  relop  5158  hash2prde  12516  symg2bas  16423 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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032
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