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Theorem eceq2 6784
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq2  |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )

Proof of Theorem eceq2
StepHypRef Expression
1 imaeq1 5089 . 2  |-  ( A  =  B  ->  ( A " { C }
)  =  ( B
" { C }
) )
2 df-ec 6749 . 2  |-  [ C ] A  =  ( A " { C }
)
3 df-ec 6749 . 2  |-  [ C ] B  =  ( B " { C }
)
41, 2, 33eqtr4g 2415 1  |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   {csn 3716   "cima 4774   [cec 6745
This theorem is referenced by:  qseq2  6797  divsval  13543  efgrelexlemb  15158  efgcpbllemb  15163  vrgpfval  15174  znzrh2  16605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-ec 6749
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