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Theorem seeq1 4365
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 3231 . . 3  |-  ( R  =  S  ->  S  C_  R )
2 sess1 4361 . . 3  |-  ( S 
C_  R  ->  ( R Se  A  ->  S Se  A
) )
31, 2syl 15 . 2  |-  ( R  =  S  ->  ( R Se  A  ->  S Se  A
) )
4 eqimss 3230 . . 3  |-  ( R  =  S  ->  R  C_  S )
5 sess1 4361 . . 3  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
64, 5syl 15 . 2  |-  ( R  =  S  ->  ( S Se  A  ->  R Se  A
) )
73, 6impbid 183 1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152   Se wse 4350
This theorem is referenced by:  oieq1  7227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-br 4024  df-se 4353
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