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Theorem 3sstr3i 3389
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 3377 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4mpbi 208 1  |-  C  C_  D
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    C_ wss 3323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3330  df-ss 3337
This theorem is referenced by:  odf1o2  16063  leordtval2  18791  uniiccvol  21035  ballotlem2  26823
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