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Theorem sseq12i 3376
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 3373 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
41, 2, 3mp2an 655 1  |-  ( A 
C_  C  <->  B  C_  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    C_ wss 3322
This theorem is referenced by:  3sstr3i  3388  3sstr4i  3389  3sstr3g  3390  3sstr4g  3391  ss2rab  3421  pjordi  23681  mdsldmd1i  23839  iuninc  24016  cvmlift2lem12  25006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336
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