Theorem eceq2 4284

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 2421	. . 3 |- (A = B -> {A} = {B})
2	1	imaeq2d 3410	. 2 |- (A = B -> (C"{A}) = (C"{B}))
3		df-ec 4269	. 2 |- [A]C = (C"{A})
4		df-ec 4269	. 2 |- [B]C = (C"{B})
5	2, 3, 4	3eqtr4g 1534	1 |- (A = B -> [A]C = [B]C)

Syntax hints:   -> wi 3   = wceq 958  {csn 2413  "cima 3179  [cec 4265

This theorem is referenced by:  erth 4288  ecelqsi 4298  snec 4302  ecoptocl 4309  brecop 4312  th3qlem1 4320  th3qlem2 4321  th3q 4323  oprec 4324  ecoprcom 4325  ecoprass 4326  ecoprdi 4327  1qec 5080  mulidpq 5081  recmulpq 5082  ltexpq 5092  halfpq 5094  prlem934a 5149  prlem934b 5150  suppsr 5234  suppsr2 5235

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-ec 4269

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