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Theorem seeq1 4851
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 3557 . . 3  |-  ( R  =  S  ->  S  C_  R )
2 sess1 4847 . . 3  |-  ( S 
C_  R  ->  ( R Se  A  ->  S Se  A
) )
31, 2syl 16 . 2  |-  ( R  =  S  ->  ( R Se  A  ->  S Se  A
) )
4 eqimss 3556 . . 3  |-  ( R  =  S  ->  R  C_  S )
5 sess1 4847 . . 3  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
64, 5syl 16 . 2  |-  ( R  =  S  ->  ( S Se  A  ->  R Se  A
) )
73, 6impbid 191 1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    C_ wss 3476   Se wse 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-br 4448  df-se 4839
This theorem is referenced by:  oieq1  7938
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