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Theorem mndoio 25461
 Description: A monoid is an internal operation. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypothesis
Ref Expression
mndoid.1
Assertion
Ref Expression
mndoio MndOp

Proof of Theorem mndoio
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndoid.1 . . . 4
21ismndo2 21028 . . 3 MndOp MndOp
3 simp1 955 . . 3
42, 3syl6bi 219 . 2 MndOp MndOp
54pm2.43i 43 1 MndOp
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696  wral 2556  wrex 2557   cxp 4703   crn 4706  wf 5267  (class class class)co 5874  MndOpcmndo 21020 This theorem is referenced by:  expus  25468 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021
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