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Theorem 3sstr4d 3202
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4d.1  |-  ( ph  ->  A  C_  B )
3sstr4d.2  |-  ( ph  ->  C  =  A )
3sstr4d.3  |-  ( ph  ->  D  =  B )
Assertion
Ref Expression
3sstr4d  |-  ( ph  ->  C  C_  D )

Proof of Theorem 3sstr4d
StepHypRef Expression
1 3sstr4d.1 . 2  |-  ( ph  ->  A  C_  B )
2 3sstr4d.2 . . 3  |-  ( ph  ->  C  =  A )
3 3sstr4d.3 . . 3  |-  ( ph  ->  D  =  B )
42, 3sseq12d 3188 . 2  |-  ( ph  ->  ( C  C_  D  <->  A 
C_  B ) )
51, 4mpbird 167 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144
This theorem is referenced by:  rdgss  6386  sucinc2  6449  oawordi  6472  nnnninf  7126  fzoss1  10173  fzoss2  10174  clsss  13657  ntrss  13658  sslm  13786  txss12  13805  metss2lem  14036  xmettxlem  14048  xmettx  14049  nnsf  14793  nninfself  14801
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