Theorem eceq1 6713

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 3677	. . 3 |- ( A = B -> { A } = { B } )
2	1	imaeq2d 5067	. 2 |- ( A = B -> ( C " { A } ) = ( C " { B } ) )
3		df-ec 6680	. 2 |- [ A ] C = ( C " { A } )
4		df-ec 6680	. 2 |- [ B ] C = ( C " { B } )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> [ A ] C = [ B ] C )

Syntax hints:   -> wi 4   = wceq 1395   {csn 3666   "cima 4721   [cec 6676

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-ec 6680

This theorem is referenced by:  eceq1d  6714  ecelqsg  6733  snec  6741  qliftfun  6762  qliftfuns  6764  qliftval  6766  ecoptocl  6767  eroveu  6771  th3qlem1  6782  th3qlem2  6783  th3q  6785  dmaddpqlem  7560  nqpi  7561  1qec  7571  nqnq0  7624  nq0nn  7625  mulnnnq0  7633  addpinq1  7647  caucvgsrlemfv  7974  caucvgsr  7985  pitonnlem1  8028  axcaucvg  8083  divsfval  13356  divsfvalg  13357  qusghm  13814  znzrhval  14605