Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cosseqi Structured version   Visualization version   GIF version

Theorem cosseqi 37809
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
StepHypRef Expression
1 cosseqi.1 . 2 𝐴 = 𝐵
2 cosseq 37808 . 2 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
31, 2ax-mp 5 1 𝐴 = ≀ 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  ccoss 37555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-br 5142  df-opab 5204  df-coss 37793
This theorem is referenced by:  br1cossinres  37829  br1cossxrnres  37830  cosscnvid  37863
  Copyright terms: Public domain W3C validator