Theorem eceq2i 8676

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 8675	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	ax-mp 5	1 ⊢ [𝐶]𝐴 = [𝐶]𝐵

Syntax hints:   = wceq 1547  [cec 8631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635

This theorem is referenced by:  ecqusaddd  19158  rngqiprnglinlem2  21285  rngqiprngimf1lem  21287  rngqiprngimf1  21293  eccnvepres3  38659  extid  38683  ecunres  38761  dfblockliftmap2  38828  dfsucmap3  38830  dfsucmap2  38831  dfpre4  38847  br2coss  38895  eldisjlem19  39280  prjspeclsp  43062