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Theorem mndid 14690
 Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
mndlem1.b
mndlem1.p
Assertion
Ref Expression
mndid
Distinct variable groups:   ,,   ,,   , ,

Proof of Theorem mndid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndlem1.b . . 3
2 mndlem1.p . . 3
31, 2ismnd 14685 . 2
43simprbi 451 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2698  wrex 2699  cfv 5447  (class class class)co 6074  cbs 13462   cplusg 13522  cmnd 14677 This theorem is referenced by:  mndideu  14691  mndidcl  14707  mndlrid  14708  prds0g  14722 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4331  ax-pow 4370 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-ov 6077  df-mnd 14683
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