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Theorem opcon2b 35217
Description: Orthocomplement contraposition law. (negcon2 10627 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 35214 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
433adant2 1162 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
51, 2opcon3b 35216 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
64, 5syld3an3 1529 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ ( ‘( 𝑌)) = ( 𝑋)))
71, 2opococ 35215 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
873adant2 1162 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
98eqeq1d 2802 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑌)) = ( 𝑋) ↔ 𝑌 = ( 𝑋)))
106, 9bitrd 271 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  cfv 6102  Basecbs 16183  occoc 16274  OPcops 35192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-nul 4984
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-dm 5323  df-iota 6065  df-fv 6110  df-ov 6882  df-oposet 35196
This theorem is referenced by:  opcon1b  35218  riotaocN  35229  glbconxN  35398
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