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Theorem seeq2 4382
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 3244 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 sess2 4378 . . 3  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  ( R Se  A  ->  R Se  B
) )
4 eqimss 3243 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 sess2 4378 . . 3  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  ( R Se  B  ->  R Se  A
) )
73, 6impbid 183 1  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    C_ wss 3165   Se wse 4366
This theorem is referenced by:  oieq2  7244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-se 4369
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