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Theorem sseq12i 3772
 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 3769 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
41, 2, 3mp2an 710 1 (𝐴𝐶𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1632   ⊆ wss 3715 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729 This theorem is referenced by:  3sstr3i  3784  3sstr4i  3785  3sstr3g  3786  3sstr4g  3787  ss2rab  3819  rabsssn  4359  issubgr  26362  pjordi  29341  mdsldmd1i  29499  iuninc  29686  cvmlift2lem12  31603  brtrclfv2  38521  nzss  39018  hoidmvle  41320  ovolval5lem3  41374  fldhmsubc  42594  fldhmsubcALTV  42612
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