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Theorem sseq12i 3034
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 3031 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
41, 2, 3mp2an 417 1  |-  ( A 
C_  C  <->  B  C_  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    C_ wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995
This theorem is referenced by:  3sstr3i  3046  3sstr4i  3047  3sstr3g  3048  3sstr4g  3049  ss2rab  3079
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