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Theorem seeq2 4167
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq2  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)

Proof of Theorem seeq2
StepHypRef Expression
1 eqimss2 3079 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 sess2 4165 . . 3  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  ( R Se  A  ->  R Se  B
) )
4 eqimss 3078 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 sess2 4165 . . 3  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
64, 5syl 14 . 2  |-  ( A  =  B  ->  ( R Se  B  ->  R Se  A
) )
73, 6impbid 127 1  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    C_ wss 2999   Se wse 4156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-se 4160
This theorem is referenced by: (None)
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