Theorem eceq2d 8788

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	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eceq2d	⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Proof of Theorem eceq2d

Step	Hyp	Ref	Expression
1		eceq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eceq2 8786	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1540  [cec 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747

This theorem is referenced by:  vrgpfval  19784  quslsm  33433  opprqusplusg  33517  opprqusmulr  33519  qsdrngi  33523  releldmqscoss  38661  aks6d1c6lem5  42178  aks5lem3a  42190  prjspeclsp  42622