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Theorem omllaw 32241
 Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b
omllaw.l
omllaw.j
omllaw.m
omllaw.o
Assertion
Ref Expression
omllaw

Proof of Theorem omllaw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5
2 omllaw.l . . . . 5
3 omllaw.j . . . . 5
4 omllaw.m . . . . 5
5 omllaw.o . . . . 5
61, 2, 3, 4, 5isoml 32236 . . . 4
76simprbi 462 . . 3
8 breq1 4397 . . . . 5
9 id 22 . . . . . . 7
10 fveq2 5848 . . . . . . . 8
1110oveq2d 6293 . . . . . . 7
129, 11oveq12d 6295 . . . . . 6
1312eqeq2d 2416 . . . . 5
148, 13imbi12d 318 . . . 4
15 breq2 4398 . . . . 5
16 id 22 . . . . . 6
17 oveq1 6284 . . . . . . 7
1817oveq2d 6293 . . . . . 6
1916, 18eqeq12d 2424 . . . . 5
2015, 19imbi12d 318 . . . 4
2114, 20rspc2v 3168 . . 3
227, 21syl5com 28 . 2
23223impib 1195 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2753   class class class wbr 4394  cfv 5568  (class class class)co 6277  cbs 14839  cple 14914  coc 14915  cjn 15895  cmee 15896  col 32172  coml 32173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-oml 32177 This theorem is referenced by:  omllaw2N  32242  omllaw3  32243  omllaw4  32244
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