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Theorem eceq2 6901
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 5157 . 2  |-  ( A  =  B  ->  ( A " { C }
)  =  ( B
" { C }
) )
2 df-ec 6866 . 2  |-  [ C ] A  =  ( A " { C }
)
3 df-ec 6866 . 2  |-  [ C ] B  =  ( B " { C }
)
41, 2, 33eqtr4g 2461 1  |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   {csn 3774   "cima 4840   [cec 6862
This theorem is referenced by:  qseq2  6914  divsval  13722  efgrelexlemb  15337  efgcpbllemb  15342  vrgpfval  15353  znzrh2  16781
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-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866
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