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Theorem ablocom 20952
 Description: An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablcom.1
Assertion
Ref Expression
ablocom

Proof of Theorem ablocom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.1 . . . . 5
21isablo 20950 . . . 4
32simprbi 450 . . 3
4 oveq1 5865 . . . . 5
5 oveq2 5866 . . . . 5
64, 5eqeq12d 2297 . . . 4
7 oveq2 5866 . . . . 5
8 oveq1 5865 . . . . 5
97, 8eqeq12d 2297 . . . 4
106, 9rspc2v 2890 . . 3
113, 10syl5com 26 . 2
12113impib 1149 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  wral 2543   crn 4690  (class class class)co 5858  cgr 20853  cablo 20948 This theorem is referenced by:  ablo32  20953  ablomuldiv  20956  ablodiv32  20959  gxdi  20963  ghablo  21036  rngocom  21059  vccom  21116  nvcom  21177  abloinvop  25353  fprodneg  25378  addvecom  25466  iscringd  26624 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861  df-ablo 20949
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