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Theorem cmbr 22155
 Description: Binary relation expressing commutes with . Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cmbr
Dummy variables are mutually distinct and distinct from all other variables.

Proof of Theorem cmbr
StepHypRef Expression
1 eleq1 2344 . . . . 5
21anbi1d 687 . . . 4
3 id 21 . . . . 5
4 ineq1 3364 . . . . . 6
5 ineq1 3364 . . . . . 6
64, 5oveq12d 5837 . . . . 5
73, 6eqeq12d 2298 . . . 4
82, 7anbi12d 693 . . 3
9 eleq1 2344 . . . . 5
109anbi2d 686 . . . 4
11 ineq2 3365 . . . . . 6
12 fveq2 5485 . . . . . . 7
1312ineq2d 3371 . . . . . 6
1411, 13oveq12d 5837 . . . . 5
1514eqeq2d 2295 . . . 4
1610, 15anbi12d 693 . . 3
17 df-cm 22154 . . 3
188, 16, 17brabg 4283 . 2
1918bianabs 852 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360   wceq 1624   wcel 1685   cin 3152   class class class wbr 4024  cfv 5221  (class class class)co 5819  cch 21501  cort 21502   chj 21505   ccm 21508 This theorem is referenced by:  cmbri  22161  cm2j  22191 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5822  df-cm 22154
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