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Theorem soeq2 4334
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 4332 . . . 4  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
2 soss 4332 . . . 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 3194 . . 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 1623    C_ wss 3152    Or wor 4313
This theorem is referenced by:  weeq2  4382  oemapso  7384  fin2i  7921  isfin2-2  7945  fin1a2lem10  8035  zorn2lem7  8129  zornn0g  8132  opsrtoslem2  16226  sltsolem1  24322  soeq12d  27134  aomclem1  27151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-in 3159  df-ss 3166  df-po 4314  df-so 4315
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