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Theorem opcon2b 38907
Description: Orthocomplement contraposition law. (negcon2 11553 analog.) (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon2b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))

Proof of Theorem opcon2b
StepHypRef Expression
1 opoccl.b . . . . 5 𝐵 = (Base‘𝐾)
2 opoccl.o . . . . 5 = (oc‘𝐾)
31, 2opoccl 38904 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
433adant2 1128 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
51, 2opcon3b 38906 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
64, 5syld3an3 1406 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
71, 2opococ 38905 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
873adant2 1128 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
98eqeq1d 2728 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑌)) = ( 𝑋) ↔ 𝑌 = ( 𝑋)))
106, 9bitrd 278 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1534  wcel 2099  cfv 6545  Basecbs 17207  occoc 17268  OPcops 38882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-dm 5684  df-iota 6497  df-fv 6553  df-ov 7418  df-oposet 38886
This theorem is referenced by:  opcon1b  38908  riotaocN  38919  glbconxN  39089
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