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Theorem isgrp 14818
 Description: The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrp.b
isgrp.p
isgrp.z
Assertion
Ref Expression
isgrp
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem isgrp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . 4
2 isgrp.b . . . 4
31, 2syl6eqr 2488 . . 3
4 fveq2 5730 . . . . . . 7
5 isgrp.p . . . . . . 7
64, 5syl6eqr 2488 . . . . . 6
76oveqd 6100 . . . . 5
8 fveq2 5730 . . . . . 6
9 isgrp.z . . . . . 6
108, 9syl6eqr 2488 . . . . 5
117, 10eqeq12d 2452 . . . 4
123, 11rexeqbidv 2919 . . 3
133, 12raleqbidv 2918 . 2
14 df-grp 14814 . 2
1513, 14elrab2 3096 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  c0g 13725  cmnd 14686  cgrp 14687 This theorem is referenced by:  grpmnd  14819  grpinvex  14822  grppropd  14825  isgrpd2e  14829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-grp 14814
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