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Theorem preq12b 4142
 Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preqr1.a
preqr1.b
preq12b.c
preq12b.d
Assertion
Ref Expression
preq12b

Proof of Theorem preq12b
StepHypRef Expression
1 preqr1.a . . . . . 6
21prid1 4071 . . . . 5
3 eleq2 2538 . . . . 5
42, 3mpbii 216 . . . 4
51elpr 3977 . . . 4
64, 5sylib 201 . . 3
7 preq1 4042 . . . . . . . 8
87eqeq1d 2473 . . . . . . 7
9 preqr1.b . . . . . . . 8
10 preq12b.d . . . . . . . 8
119, 10preqr2 4141 . . . . . . 7
128, 11syl6bi 236 . . . . . 6
1312com12 31 . . . . 5
1413ancld 562 . . . 4
15 prcom 4041 . . . . . . 7
1615eqeq2i 2483 . . . . . 6
17 preq1 4042 . . . . . . . . 9
1817eqeq1d 2473 . . . . . . . 8
19 preq12b.c . . . . . . . . 9
209, 19preqr2 4141 . . . . . . . 8
2118, 20syl6bi 236 . . . . . . 7
2221com12 31 . . . . . 6
2316, 22sylbi 200 . . . . 5
2423ancld 562 . . . 4
2514, 24orim12d 856 . . 3
266, 25mpd 15 . 2
27 preq12 4044 . . 3
28 prcom 4041 . . . . 5
2917, 28syl6eq 2521 . . . 4
30 preq1 4042 . . . 4
3129, 30sylan9eq 2525 . . 3
3227, 31jaoi 386 . 2
3326, 32impbii 192 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wo 375   wa 376   wceq 1452   wcel 1904  cvv 3031  cpr 3961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962 This theorem is referenced by:  prel12  4143  opthpr  4144  preq12bg  4146  preqsn  4151  opeqpr  4698  preleq  8140  axlowdimlem13  25063  wlkdvspthlem  25416  altopthsn  30799  upgrwlkdvdelem  39928
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