Theorem eceq1 6802

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

Step	Hyp	Ref	Expression
1		sneq 3700	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 5101	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6769	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6769	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2290	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1398   {csn 3689   "cima 4752   [cec 6765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769

This theorem is referenced by:  eceq1d  6803  ecelqsg  6822  snec  6830  qliftfun  6851  qliftfuns  6853  qliftval  6855  ecoptocl  6856  eroveu  6860  th3qlem1  6871  th3qlem2  6872  th3q  6874  dmaddpqlem  7692  nqpi  7693  1qec  7703  nqnq0  7756  nq0nn  7757  mulnnnq0  7765  addpinq1  7779  caucvgsrlemfv  8106  caucvgsr  8117  pitonnlem1  8160  axcaucvg  8215  divsfval  13541  divsfvalg  13542  qusghm  13999  znzrhval  14795