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Theorem sseq12i 3125
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 3122 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
41, 2, 3mp2an 422 1  |-  ( A 
C_  C  <->  B  C_  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  3sstr3i  3137  3sstr4i  3138  3sstr3g  3139  3sstr4g  3140  ss2rab  3173
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