Theorem eceq1i 37748

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 8763	. 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
3	1, 2	ax-mp 5	1 ⊢ [𝐴]𝐶 = [𝐵]𝐶

Syntax hints:   = wceq 1534  [cec 8723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727

This theorem is referenced by: (None)