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Theorem seeq2 4784
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 3452 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 sess2 4780 . . 3  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2syl 17 . 2  |-  ( A  =  B  ->  ( R Se  A  ->  R Se  B
) )
4 eqimss 3451 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 sess2 4780 . . 3  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
64, 5syl 17 . 2  |-  ( A  =  B  ->  ( R Se  B  ->  R Se  A
) )
73, 6impbid 195 1  |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1447    C_ wss 3371   Se wse 4768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741  df-rab 2745  df-v 3014  df-in 3378  df-ss 3385  df-se 4771
This theorem is referenced by:  oieq2  8014
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