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Theorem preq12b 4127
 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 3827 . . . . 5
3 eleq2 2414 . . . . 5
42, 3mpbii 202 . . . 4
51elpr 3751 . . . 4
64, 5sylib 188 . . 3
7 preq1 3799 . . . . . . . 8
87eqeq1d 2361 . . . . . . 7
9 preq12b.2 . . . . . . . 8
10 preq12b.4 . . . . . . . 8
119, 10preqr2 4125 . . . . . . 7
128, 11syl6bi 219 . . . . . 6
1312com12 27 . . . . 5
1413ancld 536 . . . 4
15 prcom 3798 . . . . . . 7
1615eqeq2i 2363 . . . . . 6
17 preq1 3799 . . . . . . . . 9
1817eqeq1d 2361 . . . . . . . 8
19 preq12b.3 . . . . . . . . 9
209, 19preqr2 4125 . . . . . . . 8
2118, 20syl6bi 219 . . . . . . 7
2221com12 27 . . . . . 6
2316, 22sylbi 187 . . . . 5
2423ancld 536 . . . 4
2514, 24orim12d 811 . . 3
266, 25mpd 14 . 2
27 preq12 3801 . . 3
28 prcom 3798 . . . . 5
2917, 28syl6eq 2401 . . . 4
30 preq1 3799 . . . 4
3129, 30sylan9eq 2405 . . 3
3227, 31jaoi 368 . 2
3326, 32impbii 180 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wo 357   wa 358   wceq 1642   wcel 1710  cvv 2859  cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  preq12bg  4128
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