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Theorem seeq2 4841
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq2  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)

Proof of Theorem seeq2
StepHypRef Expression
1 eqimss2 3542 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 sess2 4837 . . 3  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2syl 16 . 2  |-  ( A  =  B  ->  ( R Se  A  ->  R Se  B
) )
4 eqimss 3541 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 sess2 4837 . . 3  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
64, 5syl 16 . 2  |-  ( A  =  B  ->  ( R Se  B  ->  R Se  A
) )
73, 6impbid 191 1  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    C_ wss 3461   Se wse 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-in 3468  df-ss 3475  df-se 4828
This theorem is referenced by:  oieq2  7930
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