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Theorem opcon1b 34311
Description: Orthocomplement contraposition law. (negcon1 10330 analog.) (Contributed by NM, 24-Jan-2012.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon1b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 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 34310 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))
4 eqcom 2628 . . 3 (( 𝑌) = 𝑋𝑋 = ( 𝑌))
5 eqcom 2628 . . 3 (( 𝑋) = 𝑌𝑌 = ( 𝑋))
63, 4, 53bitr4g 303 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = 𝑋 ↔ ( 𝑋) = 𝑌))
76bicomd 213 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = 𝑌 ↔ ( 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1037   = wceq 1482  wcel 1989  cfv 5886  Basecbs 15851  occoc 15943  OPcops 34285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-dm 5122  df-iota 5849  df-fv 5894  df-ov 6650  df-oposet 34289
This theorem is referenced by:  opoc0  34316
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