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Theorem soeq2 4437
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 4435 . . . 4  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
2 soss 4435 . . . 4  |-  ( B 
C_  A  ->  ( R  Or  A  ->  R  Or  B ) )
31, 2anim12i 549 . . 3  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( R  Or  B  ->  R  Or  A
)  /\  ( R  Or  A  ->  R  Or  B ) ) )
4 eqss 3280 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 dfbi2 609 . . 3  |-  ( ( R  Or  B  <->  R  Or  A )  <->  ( ( R  Or  B  ->  R  Or  A )  /\  ( R  Or  A  ->  R  Or  B ) ) )
63, 4, 53imtr4i 257 . 2  |-  ( A  =  B  ->  ( R  Or  B  <->  R  Or  A ) )
76bicomd 192 1  |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    C_ wss 3238    Or wor 4416
This theorem is referenced by:  weeq2  4485  oemapso  7531  fin2i  8068  isfin2-2  8092  fin1a2lem10  8182  zorn2lem7  8276  zornn0g  8279  opsrtoslem2  16436  sltsolem1  25063  soeq12d  26640  aomclem1  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-in 3245  df-ss 3252  df-po 4417  df-so 4418
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