Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3sstr3i Structured version   Visualization version   GIF version

Theorem 3sstr3i 3606
 Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr3.1 𝐴𝐵
3sstr3.2 𝐴 = 𝐶
3sstr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3sstr3i 𝐶𝐷

Proof of Theorem 3sstr3i
StepHypRef Expression
1 3sstr3.1 . 2 𝐴𝐵
2 3sstr3.2 . . 3 𝐴 = 𝐶
3 3sstr3.3 . . 3 𝐵 = 𝐷
42, 3sseq12i 3594 . 2 (𝐴𝐵𝐶𝐷)
51, 4mpbi 219 1 𝐶𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by:  odf1o2  17811  leordtval2  20826  uniiccvol  23154  ballotlem2  29877  cotrcltrcl  37036
 Copyright terms: Public domain W3C validator