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Theorem abl32 15433
 Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b
ablcom.p
abl32.g
abl32.x
abl32.y
abl32.z
Assertion
Ref Expression
abl32

Proof of Theorem abl32
StepHypRef Expression
1 abl32.g . . 3
2 ablcmn 15418 . . 3 CMnd
31, 2syl 16 . 2 CMnd
4 abl32.x . 2
5 abl32.y . 2
6 abl32.z . 2
7 ablcom.b . . 3
8 ablcom.p . . 3
97, 8cmn32 15430 . 2 CMnd
103, 4, 5, 6, 9syl13anc 1186 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cfv 5454  (class class class)co 6081  cbs 13469   cplusg 13529  CMndccmn 15412  cabel 15413 This theorem is referenced by:  baerlem5alem1  32506  baerlem5blem1  32507 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 4338  ax-pow 4377 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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-mnd 14690  df-cmn 15414  df-abl 15415
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