Theorem eceq2 4285

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 2422	. . 3 |- (A = B -> {A} = {B})
2	1	imaeq2d 3411	. 2 |- (A = B -> (C"{A}) = (C"{B}))
3		df-ec 4270	. 2 |- [A]C = (C"{A})
4		df-ec 4270	. 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 2414  "cima 3180  [cec 4266

This theorem is referenced by:  erth 4289  ecelqsi 4299  snec 4303  ecoptocl 4310  brecop 4313  th3qlem1 4321  th3qlem2 4322  th3q 4324  oprec 4325  ecoprcom 4326  ecoprass 4327  ecoprdi 4328  1qec 5087  mulidpq 5088  recmulpq 5089  ltexpq 5099  halfpq 5101  prlem934a 5156  prlem934b 5157  suppsr 5241  suppsr2 5242

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 2709  ax-pow 2749  ax-pr 2786

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 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-br 2626  df-opab 2673  df-xp 3191  df-cnv 3193  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-ec 4270

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