Theorem eceq1 6584

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 3615	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 4982	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6551	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6551	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2245	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1363   {csn 3604   "cima 4641   [cec 6547

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-ec 6551

This theorem is referenced by:  eceq1d  6585  ecelqsg  6602  snec  6610  qliftfun  6631  qliftfuns  6633  qliftval  6635  ecoptocl  6636  eroveu  6640  th3qlem1  6651  th3qlem2  6652  th3q  6654  dmaddpqlem  7390  nqpi  7391  1qec  7401  nqnq0  7454  nq0nn  7455  mulnnnq0  7463  addpinq1  7477  caucvgsrlemfv  7804  caucvgsr  7815  pitonnlem1  7858  axcaucvg  7913  divsfvalg  12767