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Theorem 3sstr3i 3378
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 3366 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4mpbi 200 1  |-  C  C_  D
Colors of variables: wff set class
Syntax hints:    = wceq 1652    C_ wss 3312
This theorem is referenced by:  odf1o2  15199  leordtval2  17268  uniiccvol  19464  ballotlem2  24738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326
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