Theorem eceq2 3215

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 1816	. . 3 |- (A = B -> {A} = {B})
2		imaeq2 2603	. . 3 |- ({A} = {B} -> (C"{A}) = (C"{B}))
3	1, 2	syl 12	. 2 |- (A = B -> (C"{A}) = (C"{B}))
4		df-ec 3202	. 2 |- [A]C = (C"{A})
5		df-ec 3202	. 2 |- [B]C = (C"{B})
6	3, 4, 5	3eqtr4g 1147	1 |- (A = B -> [A]C = [B]C)

Syntax hints:   -> wi 2   = wceq 1091  {csn 1808  "cima 2413  [cec 3198

This theorem is referenced by:  erth 3219  ecelqsi 3229  snec 3232  ecoptocl 3239  brecop 3242  th3qlem1 3250  th3qlem2 3251  th3q 3253  oprec 3254  ecoprcom 3255  ecoprass 3256  ecoprdi 3257  1qec 3862  mulidpq 3863  recmulpq 3864  ltexpq 3874  halfpq 3876  prlem934a 3931  prlem934b 3932  suppsr 4016  suppsr2 4017

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202

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