Theorem eceq2i 8798

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 8797	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	ax-mp 5	1 ⊢ [𝐶]𝐴 = [𝐶]𝐵

This proof depends on syntax axioms:   = wceq 1570  [cec 8753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757

This theorem is used by:  ecqusaddd  19457  rngqiprnglinlem2  21639  rngqiprngimf1lem  21641  rngqiprngimf1  21647  eccnvepres3  39490  extid  39514  ecunres  39592  dfblockliftmap2  39659  dfsucmap3  39661  dfsucmap2  39662  dfpre4  39678  br2coss  39726  eldisjlem19  40111  prjspeclsp  43906