Theorem eceq1i 39356

Description: Equality theorem for 𝐶-coset of 𝐴 and 𝐶-coset of 𝐵, inference version. (Contributed by Peter Mazsa, 11-May-2021.)

Hypothesis

Ref	Expression
eceq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eceq1i	⊢ [𝐴]𝐶 = [𝐵]𝐶

Proof of Theorem eceq1i

Step	Hyp	Ref	Expression
1		eceq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		eceq1 8768	. 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
3	1, 2	ax-mp 5	1 ⊢ [𝐴]𝐶 = [𝐵]𝐶

This proof depends on syntax axioms:   = wceq 1569  [cec 8726

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8730

This theorem is used by: (None)