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Theorem 3sstr4i 3106
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4.1  |-  A  C_  B
3sstr4.2  |-  C  =  A
3sstr4.3  |-  D  =  B
Assertion
Ref Expression
3sstr4i  |-  C  C_  D

Proof of Theorem 3sstr4i
StepHypRef Expression
1 3sstr4.1 . 2  |-  A  C_  B
2 3sstr4.2 . . 3  |-  C  =  A
3 3sstr4.3 . . 3  |-  D  =  B
42, 3sseq12i 3093 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4mpbir 145 1  |-  C  C_  D
Colors of variables: wff set class
Syntax hints:    = wceq 1314    C_ wss 3039
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052
This theorem is referenced by:  undif2ss  3406  pwsnss  3698  iinuniss  3863  brab2a  4560  rncoss  4777  imassrn  4860  rnin  4916  inimass  4923  imadiflem  5170  imainlem  5172  ssoprab2i  5826  npsspw  7243  axresscn  7632
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