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Theorem eceq1i 35537
Description: Equality theorem for 𝐶-coset of 𝐴 and 𝐶-coset of 𝐵, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
Hypothesis
Ref Expression
eceq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
eceq1i [𝐴]𝐶 = [𝐵]𝐶

Proof of Theorem eceq1i
StepHypRef Expression
1 eceq1i.1 . 2 𝐴 = 𝐵
2 eceq1 8330 . 2 (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
31, 2ax-mp 5 1 [𝐴]𝐶 = [𝐵]𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  [cec 8290
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ec 8294
This theorem is referenced by: (None)
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