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Theorem soeq2 4820
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 4818 . . . 4  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
2 soss 4818 . . . 4  |-  ( B 
C_  A  ->  ( R  Or  A  ->  R  Or  B ) )
31, 2anim12i 566 . . 3  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( R  Or  B  ->  R  Or  A
)  /\  ( R  Or  A  ->  R  Or  B ) ) )
4 eqss 3519 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 dfbi2 628 . . 3  |-  ( ( R  Or  B  <->  R  Or  A )  <->  ( ( R  Or  B  ->  R  Or  A )  /\  ( R  Or  A  ->  R  Or  B ) ) )
63, 4, 53imtr4i 266 . 2  |-  ( A  =  B  ->  ( R  Or  B  <->  R  Or  A ) )
76bicomd 201 1  |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    C_ wss 3476    Or wor 4799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-in 3483  df-ss 3490  df-po 4800  df-so 4801
This theorem is referenced by:  weeq2  4868  wemapso2  7980  oemapso  8102  fin2i  8676  isfin2-2  8700  fin1a2lem10  8790  zorn2lem7  8883  zornn0g  8886  opsrtoslem2  17960  sltsolem1  29281  soeq12d  30814  aomclem1  30831
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