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Theorem eceq2i 8690
Description: Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
Hypothesis
Ref Expression
eceq2i.1 𝐴 = 𝐵
Assertion
Ref Expression
eceq2i [𝐶]𝐴 = [𝐶]𝐵

Proof of Theorem eceq2i
StepHypRef Expression
1 eceq2i.1 . 2 𝐴 = 𝐵
2 eceq2 8689 . 2 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
31, 2ax-mp 5 1 [𝐶]𝐴 = [𝐶]𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651
This theorem is referenced by:  eccnvepres3  36749  extid  36774  br2coss  36903  eldisjlem19  37275  prjspeclsp  40953
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