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Theorem pgpprm 15227
 Description: Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpprm pGrp

Proof of Theorem pgpprm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3
2 eqid 2436 . . 3
31, 2ispgp 15226 . 2 pGrp
43simp1bi 972 1 pGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wral 2705  wrex 2706   class class class wbr 4212  cfv 5454  (class class class)co 6081  cn0 10221  cexp 11382  cprime 13079  cbs 13469  cgrp 14685  cod 15163   pGrp cpgp 15165 This theorem is referenced by:  subgpgp  15231  pgpssslw  15248  sylow2blem3  15256  pgpfac1lem2  15633  pgpfac1lem3a  15634  pgpfac1lem3  15635  pgpfac1lem4  15636  pgpfaclem1  15639 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  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-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-xp 4884  df-iota 5418  df-fv 5462  df-ov 6084  df-pgp 15169
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