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Theorem preq12b 3974
 Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preq12b.1
preq12b.2
preq12b.3
preq12b.4
Assertion
Ref Expression
preq12b

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6
21prid1 3912 . . . . 5
3 eleq2 2497 . . . . 5
42, 3mpbii 203 . . . 4
51elpr 3832 . . . 4
64, 5sylib 189 . . 3
7 preq1 3883 . . . . . . . 8
87eqeq1d 2444 . . . . . . 7
9 preq12b.2 . . . . . . . 8
10 preq12b.4 . . . . . . . 8
119, 10preqr2 3973 . . . . . . 7
128, 11syl6bi 220 . . . . . 6
1312com12 29 . . . . 5
1413ancld 537 . . . 4
15 prcom 3882 . . . . . . 7
1615eqeq2i 2446 . . . . . 6
17 preq1 3883 . . . . . . . . 9
1817eqeq1d 2444 . . . . . . . 8
19 preq12b.3 . . . . . . . . 9
209, 19preqr2 3973 . . . . . . . 8
2118, 20syl6bi 220 . . . . . . 7
2221com12 29 . . . . . 6
2316, 22sylbi 188 . . . . 5
2423ancld 537 . . . 4
2514, 24orim12d 812 . . 3
266, 25mpd 15 . 2
27 preq12 3885 . . 3
28 prcom 3882 . . . . 5
2917, 28syl6eq 2484 . . . 4
30 preq1 3883 . . . 4
3129, 30sylan9eq 2488 . . 3
3227, 31jaoi 369 . 2
3326, 32impbii 181 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  cvv 2956  cpr 3815 This theorem is referenced by:  prel12  3975  opthpr  3976  preq12bg  3977  preqsn  3980  opeqpr  4453  preleq  7572  wlkdvspthlem  21607  altopthsn  25806  axlowdimlem13  25893 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821
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