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Theorem opcon1b 30010
Description: Orthocomplement contraposition law. (negcon1 9115 analog.) (Contributed by NM, 24-Jan-2012.)
Hypotheses
Ref Expression
opoccl.b  |-  B  =  ( Base `  K
)
opoccl.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
opcon1b  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  =  Y  <->  (  ._|_  `  Y )  =  X ) )

Proof of Theorem opcon1b
StepHypRef Expression
1 opoccl.b . . . 4  |-  B  =  ( Base `  K
)
2 opoccl.o . . . 4  |-  ._|_  =  ( oc `  K )
31, 2opcon2b 30009 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( 
._|_  `  Y )  <->  Y  =  (  ._|_  `  X )
) )
4 eqcom 2298 . . 3  |-  ( ( 
._|_  `  Y )  =  X  <->  X  =  (  ._|_  `  Y ) )
5 eqcom 2298 . . 3  |-  ( ( 
._|_  `  X )  =  Y  <->  Y  =  (  ._|_  `  X ) )
63, 4, 53bitr4g 279 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  Y
)  =  X  <->  (  ._|_  `  X )  =  Y ) )
76bicomd 192 1  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  =  Y  <->  (  ._|_  `  Y )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271   Basecbs 13164   occoc 13232   OPcops 29984
This theorem is referenced by:  opoc0  30015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oposet 29988
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