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Theorem seeq1 4811
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 3471 . . 3  |-  ( R  =  S  ->  S  C_  R )
2 sess1 4807 . . 3  |-  ( S 
C_  R  ->  ( R Se  A  ->  S Se  A
) )
31, 2syl 17 . 2  |-  ( R  =  S  ->  ( R Se  A  ->  S Se  A
) )
4 eqimss 3470 . . 3  |-  ( R  =  S  ->  R  C_  S )
5 sess1 4807 . . 3  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
64, 5syl 17 . 2  |-  ( R  =  S  ->  ( S Se  A  ->  R Se  A
) )
73, 6impbid 195 1  |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    C_ wss 3390   Se wse 4796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-in 3397  df-ss 3404  df-br 4396  df-se 4799
This theorem is referenced by:  oieq1  8045
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