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Theorem eceq1 7398
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 4003 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 5179 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 7364 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 7364 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2486 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {csn 3993   "cima 4848   [cec 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ec 7364
This theorem is referenced by:  eceq1d  7399  ecelqsg  7417  snec  7425  qliftfun  7447  qliftfuns  7449  qliftval  7451  ecoptocl  7452  eroveu  7457  erov  7459  divsfval  15397  qusghm  16863  sylow1lem3  17180  efgi2  17303  frgpup3lem  17355  znzrhval  19041  qustgpopn  21058  qustgplem  21059  elpi1i  21983  pi1xfrf  21990  pi1xfrval  21991  pi1xfrcnvlem  21993  pi1cof  21996  pi1coval  21997  vitalilem3  22462  prtlem9  32173  prtlem11  32175
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