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Theorem soeq2 4483
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 4481 . . . 4  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
2 soss 4481 . . . 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 3323 . . 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 1649    C_ wss 3280    Or wor 4462
This theorem is referenced by:  weeq2  4531  oemapso  7594  fin2i  8131  isfin2-2  8155  fin1a2lem10  8245  zorn2lem7  8338  zornn0g  8341  opsrtoslem2  16500  sltsolem1  25536  soeq12d  27002  aomclem1  27019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-in 3287  df-ss 3294  df-po 4463  df-so 4464
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