Theorem eceq2 4262

Description: Equality theorem for equivalence class.

Assertion

Ref	Expression
eceq2	|- (A = B -> [A]C = [B]C)

Proof of Theorem eceq2

Step	Hyp	Ref	Expression
1		sneq 2407	. . 3 |- (A = B -> {A} = {B})
2	1	imaeq2d 3388	. 2 |- (A = B -> (C"{A}) = (C"{B}))
3		df-ec 4247	. 2 |- [A]C = (C"{A})
4		df-ec 4247	. 2 |- [B]C = (C"{B})
5	2, 3, 4	3eqtr4g 1523	1 |- (A = B -> [A]C = [B]C)

Syntax hints:   -> wi 3   = wceq 953  {csn 2399  "cima 3163  [cec 4243

This theorem is referenced by:  erth 4266  ecelqsi 4276  snec 4280  ecoptocl 4287  brecop 4290  th3qlem1 4298  th3qlem2 4299  th3q 4301  oprec 4302  ecoprcom 4303  ecoprass 4304  ecoprdi 4305  1qec 5040  mulidpq 5041  recmulpq 5042  ltexpq 5052  halfpq 5054  prlem934a 5109  prlem934b 5110  suppsr 5194  suppsr2 5195

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-ec 4247

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