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Theorem eceq1i 35414
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 8316 . 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 1528  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by: (None)
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