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Theorem opcon1b 38702
Description: Orthocomplement contraposition law. (negcon1 11550 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 38701 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))
4 eqcom 2735 . . 3 (( 𝑌) = 𝑋𝑋 = ( 𝑌))
5 eqcom 2735 . . 3 (( 𝑋) = 𝑌𝑌 = ( 𝑋))
63, 4, 53bitr4g 313 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = 𝑋 ↔ ( 𝑋) = 𝑌))
76bicomd 222 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = 𝑌 ↔ ( 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  cfv 6553  Basecbs 17187  occoc 17248  OPcops 38676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561  df-ov 7429  df-oposet 38680
This theorem is referenced by:  opoc0  38707
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