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Theorem ecelqsg 6918
 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 2404 . . 3
2 eceq1 6900 . . . . 5
32eqeq2d 2415 . . . 4
43rspcev 3012 . . 3
51, 4mpan2 653 . 2
6 ecexg 6868 . . . 4
7 elqsg 6915 . . . 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 1649   wcel 1721  wrex 2667  cvv 2916  cec 6862  cqs 6863 This theorem is referenced by:  ecelqsi  6919  qliftlem  6944  erov  6960  eroprf  6961  sylow2a  15208  sylow2blem1  15209  sylow2blem2  15210  cldsubg  18093 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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