Theorem eceq1i 38234

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 8804	. 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
3	1, 2	ax-mp 5	1 ⊢ [𝐴]𝐶 = [𝐵]𝐶

Syntax hints:   = wceq 1537  [cec 8763

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767

This theorem is referenced by: (None)