Theorem eceq2i 8680

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

Step	Hyp	Ref	Expression
1		eceq2i.1	. 2 ⊢ 𝐴 = 𝐵
2		eceq2 8679	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	ax-mp 5	1 ⊢ [𝐶]𝐴 = [𝐶]𝐵

Syntax hints:   = wceq 1542  [cec 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639

This theorem is referenced by:  ecqusaddd  19161  rngqiprnglinlem2  21285  rngqiprngimf1lem  21287  rngqiprngimf1  21293  eccnvepres3  38630  extid  38654  ecunres  38732  dfblockliftmap2  38799  dfsucmap3  38801  dfsucmap2  38802  dfpre4  38818  br2coss  38866  eldisjlem19  39251  prjspeclsp  43062