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Theorem 3sstr3i 3107
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
3sstr3.1  |-  A  C_  B
3sstr3.2  |-  A  =  C
3sstr3.3  |-  B  =  D
Assertion
Ref Expression
3sstr3i  |-  C  C_  D

Proof of Theorem 3sstr3i
StepHypRef Expression
1 3sstr3.1 . 2  |-  A  C_  B
2 3sstr3.2 . . 3  |-  A  =  C
3 3sstr3.3 . . 3  |-  B  =  D
42, 3sseq12i 3095 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4mpbi 144 1  |-  C  C_  D
Colors of variables: wff set class
Syntax hints:    = wceq 1316    C_ wss 3041
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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