Theorem cosseqi 35671

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 35670	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ≀ 𝐴 = ≀ 𝐵

Syntax hints:   = wceq 1533   ≀ ccoss 35452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-br 5066  df-opab 5128  df-coss 35658

This theorem is referenced by:  br1cossinres  35686  br1cossxrnres  35687  cosscnvid  35720