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Theorem seeq1 4522
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)

Proof of Theorem seeq1
StepHypRef Expression
1 eqimss2 3369 . . 3  |-  ( R  =  S  ->  S  C_  R )
2 sess1 4518 . . 3  |-  ( S 
C_  R  ->  ( R Se  A  ->  S Se  A
) )
31, 2syl 16 . 2  |-  ( R  =  S  ->  ( R Se  A  ->  S Se  A
) )
4 eqimss 3368 . . 3  |-  ( R  =  S  ->  R  C_  S )
5 sess1 4518 . . 3  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
64, 5syl 16 . 2  |-  ( R  =  S  ->  ( S Se  A  ->  R Se  A
) )
73, 6impbid 184 1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    C_ wss 3288   Se wse 4507
This theorem is referenced by:  oieq1  7445
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 2393  ax-sep 4298
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-in 3295  df-ss 3302  df-br 4181  df-se 4510
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