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

Theorem isablo 25856
 Description: The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
isabl.1
Assertion
Ref Expression
isablo
Distinct variable groups:   ,,   ,,

Proof of Theorem isablo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rneq 5080 . . . . 5
2 isabl.1 . . . . 5
31, 2syl6eqr 2488 . . . 4
4 raleq 3032 . . . . 5
54raleqbi1dv 3040 . . . 4
63, 5syl 17 . . 3
7 oveq 6311 . . . . 5
8 oveq 6311 . . . . 5
97, 8eqeq12d 2451 . . . 4
1092ralbidv 2876 . . 3
116, 10bitrd 256 . 2
12 df-ablo 25855 . 2
1311, 12elrab2 3237 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wa 370   wceq 1437   wcel 1870  wral 2782   crn 4855  (class class class)co 6305  cgr 25759  cablo 25854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-cnv 4862  df-dm 4864  df-rn 4865  df-iota 5565  df-fv 5609  df-ov 6308  df-ablo 25855 This theorem is referenced by:  ablogrpo  25857  ablocom  25858  isabloi  25861  isabloda  25872  subgoablo  25884  ghabloOLD  25942
 Copyright terms: Public domain W3C validator