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Theorem sseq12i 3290
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  |-  ( A 
C_  C  <->  B  C_  D
)

Proof of Theorem sseq12i
StepHypRef Expression
1 sseq1i.1 . 2  |-  A  =  B
2 sseq12i.2 . 2  |-  C  =  D
3 sseq12 3287 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
41, 2, 3mp2an 653 1  |-  ( A 
C_  C  <->  B  C_  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1647    C_ wss 3238
This theorem is referenced by:  3sstr3i  3302  3sstr4i  3303  3sstr3g  3304  3sstr4g  3305  ss2rab  3335  pjordi  23187  mdsldmd1i  23345  iuninc  23530  ballotlem2  24315  cvmlift2lem12  24569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-in 3245  df-ss 3252
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