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Theorem eceq1d 8742
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 8741 . 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 1542  [cec 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705
This theorem is referenced by:  brecop  8804  eroveu  8806  erov  8808  ecovcom  8817  ecovass  8818  ecovdi  8819  addsrmo  11068  mulsrmo  11069  addsrpr  11070  mulsrpr  11071  supsrlem  11106  supsr  11107  qus0  19068  qusinv  19069  qussub  19070  sylow2blem2  19489  frgpadd  19631  vrgpval  19635  vrgpinv  19637  frgpup3lem  19645  qusabl  19733  quscrng  20878  qustgplem  23625  pi1addval  24564  pi1xfrf  24569  pi1xfrval  24570  pi1xfrcnvlem  24572  pi1xfrcnv  24573  pi1cof  24575  pi1coval  24576  pi1coghm  24577  vitalilem3  25127  opprqusmulr  32605  ismntoplly  33005  linedegen  35115  fvline  35116  pzriprnglem11  46815  pzriprnglem12  46816
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