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Theorem eceq2 6945
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 5201 . 2  |-  ( A  =  B  ->  ( A " { C }
)  =  ( B
" { C }
) )
2 df-ec 6910 . 2  |-  [ C ] A  =  ( A " { C }
)
3 df-ec 6910 . 2  |-  [ C ] B  =  ( B " { C }
)
41, 2, 33eqtr4g 2495 1  |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   {csn 3816   "cima 4884   [cec 6906
This theorem is referenced by:  qseq2  6958  divsval  13772  efgrelexlemb  15387  efgcpbllemb  15392  vrgpfval  15403  znzrh2  16831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910
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