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Theorem eceq2d 8742
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 8740 . 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 1533  [cec 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702
This theorem is referenced by:  vrgpfval  19682  quslsm  33012  opprqusplusg  33099  opprqusmulr  33101  qsdrngi  33105  releldmqscoss  38034  prjspeclsp  41906
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