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Theorem soeq2 4523
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq2  |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )

Proof of Theorem soeq2
StepHypRef Expression
1 soss 4521 . . . 4  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
2 soss 4521 . . . 4  |-  ( B 
C_  A  ->  ( R  Or  A  ->  R  Or  B ) )
31, 2anim12i 550 . . 3  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( R  Or  B  ->  R  Or  A
)  /\  ( R  Or  A  ->  R  Or  B ) ) )
4 eqss 3363 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 dfbi2 610 . . 3  |-  ( ( R  Or  B  <->  R  Or  A )  <->  ( ( R  Or  B  ->  R  Or  A )  /\  ( R  Or  A  ->  R  Or  B ) ) )
63, 4, 53imtr4i 258 . 2  |-  ( A  =  B  ->  ( R  Or  B  <->  R  Or  A ) )
76bicomd 193 1  |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    C_ wss 3320    Or wor 4502
This theorem is referenced by:  weeq2  4571  oemapso  7638  fin2i  8175  isfin2-2  8199  fin1a2lem10  8289  zorn2lem7  8382  zornn0g  8385  opsrtoslem2  16545  sltsolem1  25623  soeq12d  27112  aomclem1  27129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-in 3327  df-ss 3334  df-po 4503  df-so 4504
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