MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seeq1 Structured version   Visualization version   GIF version

Theorem seeq1 5495
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))

Proof of Theorem seeq1
StepHypRef Expression
1 eqimss2 3975 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 5491 . . 3 (𝑆𝑅 → (𝑅 Se 𝐴𝑆 Se 𝐴))
31, 2syl 17 . 2 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 eqimss 3974 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 5491 . . 3 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝑅 = 𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
73, 6impbid 215 1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wss 3884   Se wse 5480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-br 5034  df-se 5483
This theorem is referenced by:  oieq1  8964
  Copyright terms: Public domain W3C validator