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Theorem eceq1 6712
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 3664 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 5028 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 6678 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 6678 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2353 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {csn 3653   "cima 4708   [cec 6674
This theorem is referenced by:  ecelqsg  6730  snec  6738  qliftfun  6759  qliftfuns  6761  qliftval  6763  ecoptocl  6764  brecop  6767  eroveu  6769  erov  6771  th3qlem1  6780  th3qlem2  6781  th3q  6783  ovec  6784  ecovcom  6785  ecovass  6786  ecovdi  6787  supsrlem  8749  supsr  8750  divsfval  13465  divs0  14691  divsinv  14692  divssub  14693  divsghm  14735  sylow1lem3  14927  sylow2blem2  14948  efgi2  15050  frgpadd  15088  vrgpval  15092  vrgpinv  15094  frgpup3lem  15102  divsabl  15173  divscrng  16008  znzrhval  16516  divstgpopn  17818  divstgplem  17819  elpi1i  18560  pi1addval  18562  pi1xfrf  18567  pi1xfrval  18568  pi1xfrcnvlem  18570  pi1xfrcnv  18571  pi1cof  18573  pi1coval  18574  pi1coghm  18575  vitalilem3  18981  linedegen  24838  fvline  24839  pdiveql  26271  prtlem9  26835  prtlem11  26837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678
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