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Theorem seeq1 4806
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 3485 . . 3  |-  ( R  =  S  ->  S  C_  R )
2 sess1 4802 . . 3  |-  ( S 
C_  R  ->  ( R Se  A  ->  S Se  A
) )
31, 2syl 17 . 2  |-  ( R  =  S  ->  ( R Se  A  ->  S Se  A
) )
4 eqimss 3484 . . 3  |-  ( R  =  S  ->  R  C_  S )
5 sess1 4802 . . 3  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
64, 5syl 17 . 2  |-  ( R  =  S  ->  ( S Se  A  ->  R Se  A
) )
73, 6impbid 194 1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444    C_ wss 3404   Se wse 4791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-v 3047  df-in 3411  df-ss 3418  df-br 4403  df-se 4794
This theorem is referenced by:  oieq1  8027
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