MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prel12 Unicode version

Theorem prel12 4207
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 4138 . . . 4
3 eleq2 2530 . . . 4
42, 3mpbii 211 . . 3
5 preq12b.2 . . . . 5
65prid2 4139 . . . 4
7 eleq2 2530 . . . 4
86, 7mpbii 211 . . 3
94, 8jca 532 . 2
101elpr 4047 . . . 4
11 eqeq2 2472 . . . . . . . . . . . 12
1211notbid 294 . . . . . . . . . . 11
13 orel2 383 . . . . . . . . . . 11
1412, 13syl6bi 228 . . . . . . . . . 10
1514com3l 81 . . . . . . . . 9
1615imp 429 . . . . . . . 8
1716ancrd 554 . . . . . . 7
18 eqeq2 2472 . . . . . . . . . . . 12
1918notbid 294 . . . . . . . . . . 11
20 orel1 382 . . . . . . . . . . 11
2119, 20syl6bi 228 . . . . . . . . . 10
2221com3l 81 . . . . . . . . 9
2322imp 429 . . . . . . . 8
2423ancrd 554 . . . . . . 7
2517, 24orim12d 838 . . . . . 6
265elpr 4047 . . . . . . 7
27 orcom 387 . . . . . . 7
2826, 27bitri 249 . . . . . 6
29 preq12b.3 . . . . . . 7
30 preq12b.4 . . . . . . 7
311, 5, 29, 30preq12b 4206 . . . . . 6
3225, 28, 313imtr4g 270 . . . . 5
3332ex 434 . . . 4
3410, 33syl5bi 217 . . 3
3534impd 431 . 2
369, 35impbid2 204 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  {cpr 4031
This theorem is referenced by:  prel12g  4210  dfac2  8532
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
  Copyright terms: Public domain W3C validator