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Theorem eceq2 6697
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 5007 . 2  |-  ( A  =  B  ->  ( A " { C }
)  =  ( B
" { C }
) )
2 df-ec 6662 . 2  |-  [ C ] A  =  ( A " { C }
)
3 df-ec 6662 . 2  |-  [ C ] B  =  ( B " { C }
)
41, 2, 33eqtr4g 2340 1  |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {csn 3640   "cima 4692   [cec 6658
This theorem is referenced by:  qseq2  6710  divsval  13444  efgrelexlemb  15059  efgcpbllemb  15064  vrgpfval  15075  znzrh2  16499  pdiveql  26168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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