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Theorem eceq2d 8787
Description: Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
Hypothesis
Ref Expression
eceq2d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eceq2d (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Proof of Theorem eceq2d
StepHypRef Expression
1 eceq2d.1 . 2 (𝜑𝐴 = 𝐵)
2 eceq2 8785 . 2 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
31, 2syl 17 1 (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746
This theorem is referenced by:  vrgpfval  19799  quslsm  33413  opprqusplusg  33497  opprqusmulr  33499  qsdrngi  33503  releldmqscoss  38642  aks6d1c6lem5  42159  aks5lem3a  42171  prjspeclsp  42599
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