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Theorem ecelqsg 6951
 Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecelqsg

Proof of Theorem ecelqsg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3
2 eceq1 6933 . . . . 5
32eqeq2d 2446 . . . 4
43rspcev 3044 . . 3
51, 4mpan2 653 . 2
6 ecexg 6901 . . . 4
7 elqsg 6948 . . . 4
86, 7syl 16 . . 3
98biimpar 472 . 2
105, 9sylan2 461 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wrex 2698  cvv 2948  cec 6895  cqs 6896 This theorem is referenced by:  ecelqsi  6952  qliftlem  6977  erov  6993  eroprf  6994  sylow2a  15245  sylow2blem1  15246  sylow2blem2  15247  cldsubg  18132 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-ec 6899  df-qs 6903
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