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Theorem isablo 20966
 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 4920 . . . . 5
2 isabl.1 . . . . 5
31, 2syl6eqr 2346 . . . 4
4 raleq 2749 . . . . 5
54raleqbi1dv 2757 . . . 4
63, 5syl 15 . . 3
7 oveq 5880 . . . . 5
8 oveq 5880 . . . . 5
97, 8eqeq12d 2310 . . . 4
1092ralbidv 2598 . . 3
116, 10bitrd 244 . 2
12 df-ablo 20965 . 2
1311, 12elrab2 2938 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wceq 1632   wcel 1696  wral 2556   crn 4706  (class class class)co 5874  cgr 20869  cablo 20964 This theorem is referenced by:  ablogrpo  20967  ablocom  20968  isabloi  20971  isabloda  20982  subgoablo  20994  ghablo  21052  ablocomgrp  25445  tcnvec  25793 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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-ablo 20965
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