MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3sstr3g Structured version   Visualization version   GIF version

Theorem 3sstr3g 4026
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3g.1 (𝜑𝐴𝐵)
3sstr3g.2 𝐴 = 𝐶
3sstr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3sstr3g (𝜑𝐶𝐷)

Proof of Theorem 3sstr3g
StepHypRef Expression
1 3sstr3g.1 . 2 (𝜑𝐴𝐵)
2 3sstr3g.2 . . 3 𝐴 = 𝐶
3 3sstr3g.3 . . 3 𝐵 = 𝐷
42, 3sseq12i 4012 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 217 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wss 3949
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966
This theorem is referenced by:  complss  4147  uniintsn  4994  fpwwe2lem12  10673  hmeocls  23692  hmeontr  23693  usgrumgruspgr  29015  chsscon3i  31291  pjss1coi  31993  mdslmd2i  32160  satffunlem2lem2  35049  ssbnd  37294  bnd2lem  37297  trclubgNEW  43079  nzss  43785
  Copyright terms: Public domain W3C validator