Theorem eceq1 6832

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 3716	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 5121	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6799	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6799	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1402   {csn 3705   "cima 4772   [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799

This theorem is referenced by:  eceq1d  6833  ecelqsg  6852  snec  6860  qliftfun  6881  qliftfuns  6883  qliftval  6885  ecoptocl  6886  eroveu  6890  th3qlem1  6901  th3qlem2  6902  th3q  6904  dmaddpqlem  7734  nqpi  7735  1qec  7745  nqnq0  7798  nq0nn  7799  mulnnnq0  7807  addpinq1  7821  caucvgsrlemfv  8148  caucvgsr  8159  pitonnlem1  8202  axcaucvg  8257  divsfval  13626  divsfvalg  13627  qusghm  14062  znzrhval  14954