Theorem cosseqi 39021

Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)

Hypothesis

Ref	Expression
cosseqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
cosseqi	⊢ ≀ 𝐴 = ≀ 𝐵

Proof of Theorem cosseqi

Step	Hyp	Ref	Expression
1		cosseqi.1	. 2 ⊢ 𝐴 = 𝐵
2		cosseq 39020	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ≀ 𝐴 = ≀ 𝐵

Syntax hints:   = wceq 1562   ≀ ccoss 38687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-br 5103  df-opab 5165  df-coss 39005

This theorem is referenced by:  br1cossinres  39041  br1cossxrnres  39042  cosscnvid  39075