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Theorem eceq1 6900
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )

Proof of Theorem eceq1
StepHypRef Expression
1 sneq 3785 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 5162 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 6866 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 6866 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2461 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   {csn 3774   "cima 4840   [cec 6862
This theorem is referenced by:  ecelqsg  6918  snec  6926  qliftfun  6948  qliftfuns  6950  qliftval  6952  ecoptocl  6953  brecop  6956  eroveu  6958  erov  6960  th3qlem1  6969  th3qlem2  6970  th3q  6972  ovec  6973  ecovcom  6974  ecovass  6975  ecovdi  6976  supsrlem  8942  supsr  8943  divsfval  13727  divs0  14953  divsinv  14954  divssub  14955  divsghm  14997  sylow1lem3  15189  sylow2blem2  15210  efgi2  15312  frgpadd  15350  vrgpval  15354  vrgpinv  15356  frgpup3lem  15364  divsabl  15435  divscrng  16266  znzrhval  16782  divstgpopn  18102  divstgplem  18103  elpi1i  19024  pi1addval  19026  pi1xfrf  19031  pi1xfrval  19032  pi1xfrcnvlem  19034  pi1xfrcnv  19035  pi1cof  19037  pi1coval  19038  pi1coghm  19039  vitalilem3  19455  linedegen  25981  fvline  25982  prtlem9  26603  prtlem11  26605
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  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-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
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