Theorem eceq2 4278

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 2417	. . 3 |- (A = B -> {A} = {B})
2	1	imaeq2d 3404	. 2 |- (A = B -> (C"{A}) = (C"{B}))
3		df-ec 4263	. 2 |- [A]C = (C"{A})
4		df-ec 4263	. 2 |- [B]C = (C"{B})
5	2, 3, 4	3eqtr4g 1531	1 |- (A = B -> [A]C = [B]C)

Syntax hints:   -> wi 3   = wceq 956  {csn 2409  "cima 3173  [cec 4259

This theorem is referenced by:  erth 4282  ecelqsi 4292  snec 4296  ecoptocl 4303  brecop 4306  th3qlem1 4314  th3qlem2 4315  th3q 4317  oprec 4318  ecoprcom 4319  ecoprass 4320  ecoprdi 4321  1qec 5068  mulidpq 5069  recmulpq 5070  ltexpq 5080  halfpq 5082  prlem934a 5137  prlem934b 5138  suppsr 5222  suppsr2 5223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-ec 4263

Copyright terms: Public domain