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Theorem isgrpod 21886
 Description: Properties that determine a group operation. (Renamed from isgrpd 14830 to isgrpod 21886 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpda.1
isgrpda.2
isgrpda.3
isgrpda.4
isgrpda.5
isgrpod.6
isgrpod.7
Assertion
Ref Expression
isgrpod
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem isgrpod
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isgrpda.1 . 2
2 isgrpda.2 . 2
3 isgrpda.3 . 2
4 isgrpda.4 . 2
5 isgrpda.5 . 2
6 isgrpod.6 . . 3
7 isgrpod.7 . . 3
8 oveq1 6088 . . . . 5
98eqeq1d 2444 . . . 4
109rspcev 3052 . . 3
116, 7, 10syl2anc 643 . 2
121, 2, 3, 4, 5, 11isgrpda 21885 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wrex 2706  cvv 2956   cxp 4876  wf 5450  (class class class)co 6081  cgr 21774 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-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  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-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-grpo 21779
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