Theorem eceq2i 8736

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 8735	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	ax-mp 5	1 ⊢ [𝐶]𝐴 = [𝐶]𝐵

Syntax hints:   = wceq 1567  [cec 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8695

This theorem is referenced by:  ecqusaddd  19262  rngqiprnglinlem2  21402  rngqiprngimf1lem  21404  rngqiprngimf1  21410  eccnvepres3  38830  extid  38854  ecunres  38932  dfblockliftmap2  38999  dfsucmap3  39001  dfsucmap2  39002  dfpre4  39018  br2coss  39066  eldisjlem19  39451  prjspeclsp  43235