MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sseq12i Structured version   Visualization version   GIF version

Theorem sseq12i 3593
Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
sseq1i.1 𝐴 = 𝐵
sseq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
sseq12i (𝐴𝐶𝐵𝐷)

Proof of Theorem sseq12i
StepHypRef Expression
1 sseq1i.1 . 2 𝐴 = 𝐵
2 sseq12i.2 . 2 𝐶 = 𝐷
3 sseq12 3590 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
41, 2, 3mp2an 703 1 (𝐴𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553
This theorem is referenced by:  3sstr3i  3605  3sstr4i  3606  3sstr3g  3607  3sstr4g  3608  ss2rab  3640  rabsssn  4161  pjordi  28222  mdsldmd1i  28380  iuninc  28567  cvmlift2lem12  30356  brtrclfv2  36841  nzss  37341  hoidmvle  39294  ovolval5lem3  39348  issubgr  40497  fldhmsubc  41878  fldhmsubcALTV  41897
  Copyright terms: Public domain W3C validator