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Theorem eceq1d 8783
Description: Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
Hypothesis
Ref Expression
eceq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eceq1d (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)

Proof of Theorem eceq1d
StepHypRef Expression
1 eceq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 eceq1 8782 . 2 (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
31, 2syl 17 1 (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  [cec 8741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745
This theorem is referenced by:  brecop  8848  eroveu  8850  erov  8852  ecovcom  8861  ecovass  8862  ecovdi  8863  addsrmo  11110  mulsrmo  11111  addsrpr  11112  mulsrpr  11113  supsrlem  11148  supsr  11149  qus0  19219  qusinv  19220  qussub  19221  sylow2blem2  19653  frgpadd  19795  vrgpval  19799  vrgpinv  19801  frgpup3lem  19809  qusabl  19897  quscrng  21310  pzriprnglem11  21519  pzriprnglem12  21520  qustgplem  24144  pi1addval  25094  pi1xfrf  25099  pi1xfrval  25100  pi1xfrcnvlem  25102  pi1xfrcnv  25103  pi1cof  25105  pi1coval  25106  pi1coghm  25107  vitalilem3  25658  elrlocbasi  33252  rlocaddval  33254  rlocmulval  33255  rloccring  33256  rloc0g  33257  rloc1r  33258  rlocf1  33259  idomsubr  33290  opprqusmulr  33498  zringfrac  33561  ismntoplly  33987  linedegen  36124  fvline  36125  aks5lem3a  42170  aks5lem5a  42172  aks5lem6  42173
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