Theorem cosseqi 37899

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 37898	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ≀ 𝐴 = ≀ 𝐵

Syntax hints:   = wceq 1534   ≀ ccoss 37648

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-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-br 5149  df-opab 5211  df-coss 37883

This theorem is referenced by:  br1cossinres  37919  br1cossxrnres  37920  cosscnvid  37953