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Theorem eceq2d 34585
 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 8049 . 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 1656  [cec 8007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ec 8011 This theorem is referenced by: (None)
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