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Theorem grpprop 14824
 Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpprop.b
grpprop.p
Assertion
Ref Expression
grpprop

Proof of Theorem grpprop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2437 . . 3
2 grpprop.b . . . 4
32a1i 11 . . 3
4 grpprop.p . . . . 5
54oveqi 6094 . . . 4
65a1i 11 . . 3
71, 3, 6grppropd 14823 . 2
87trud 1332 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wtru 1325   wceq 1652   wcel 1725  cfv 5454  (class class class)co 6081  cbs 13469   cplusg 13529  cgrp 14685 This theorem is referenced by:  grppropstr  14825  grpss  14826  opprrng  15736  opprsubg  15741 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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 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-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-0g 13727  df-mnd 14690  df-grp 14812
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