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Theorem seeq2 4230
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 3120 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 sess2 4228 . . 3  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  ( R Se  A  ->  R Se  B
) )
4 eqimss 3119 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 sess2 4228 . . 3  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
64, 5syl 14 . 2  |-  ( A  =  B  ->  ( R Se  B  ->  R Se  A
) )
73, 6impbid 128 1  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314    C_ wss 3039   Se wse 4219
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-se 4223
This theorem is referenced by: (None)
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