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Theorem prel12 3967
 Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preq12b.1
preq12b.2
preq12b.3
preq12b.4
Assertion
Ref Expression
prel12

Proof of Theorem prel12
StepHypRef Expression
1 preq12b.1 . . . . 5
21prid1 3904 . . . 4
3 eleq2 2496 . . . 4
42, 3mpbii 203 . . 3
5 preq12b.2 . . . . 5
65prid2 3905 . . . 4
7 eleq2 2496 . . . 4
86, 7mpbii 203 . . 3
94, 8jca 519 . 2
101elpr 3824 . . . 4
11 eqeq2 2444 . . . . . . . . . . . 12
1211notbid 286 . . . . . . . . . . 11
13 orel2 373 . . . . . . . . . . 11
1412, 13syl6bi 220 . . . . . . . . . 10
1514com3l 77 . . . . . . . . 9
1615imp 419 . . . . . . . 8
1716ancrd 538 . . . . . . 7
18 eqeq2 2444 . . . . . . . . . . . 12
1918notbid 286 . . . . . . . . . . 11
20 orel1 372 . . . . . . . . . . 11
2119, 20syl6bi 220 . . . . . . . . . 10
2221com3l 77 . . . . . . . . 9
2322imp 419 . . . . . . . 8
2423ancrd 538 . . . . . . 7
2517, 24orim12d 812 . . . . . 6
265elpr 3824 . . . . . . 7
27 orcom 377 . . . . . . 7
2826, 27bitri 241 . . . . . 6
29 preq12b.3 . . . . . . 7
30 preq12b.4 . . . . . . 7
311, 5, 29, 30preq12b 3966 . . . . . 6
3225, 28, 313imtr4g 262 . . . . 5
3332ex 424 . . . 4
3410, 33syl5bi 209 . . 3
3534imp3a 421 . 2
369, 35impbid2 196 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  cvv 2948  cpr 3807 This theorem is referenced by:  dfac2  8003 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 2416 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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813
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