Theorem eceq1i 38270

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 8789	. 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
3	1, 2	ax-mp 5	1 ⊢ [𝐴]𝐶 = [𝐵]𝐶

Syntax hints:   = wceq 1538  [cec 8748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-xp 5696  df-cnv 5698  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-ec 8752

This theorem is referenced by: (None)