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Theorem sseq12i 3334
Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
sseq1i.1  |-  A  =  B
sseq12i.2  |-  C  =  D
Assertion
Ref Expression
sseq12i  |-  ( A 
C_  C  <->  B  C_  D
)

Proof of Theorem sseq12i
StepHypRef Expression
1 sseq1i.1 . 2  |-  A  =  B
2 sseq12i.2 . 2  |-  C  =  D
3 sseq12 3331 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
41, 2, 3mp2an 654 1  |-  ( A 
C_  C  <->  B  C_  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    C_ wss 3280
This theorem is referenced by:  3sstr3i  3346  3sstr4i  3347  3sstr3g  3348  3sstr4g  3349  ss2rab  3379  pjordi  23629  mdsldmd1i  23787  iuninc  23964  cvmlift2lem12  24954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-in 3287  df-ss 3294
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