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Theorem opcon3b 38700
Description: Contraposition law for orthoposets. (chcon3i 31296 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon3b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Proof of Theorem opcon3b
StepHypRef Expression
1 fveq2 6902 . . 3 (𝑌 = 𝑋 → ( 𝑌) = ( 𝑋))
21eqcoms 2736 . 2 (𝑋 = 𝑌 → ( 𝑌) = ( 𝑋))
3 fveq2 6902 . . . 4 (( 𝑋) = ( 𝑌) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
43eqcoms 2736 . . 3 (( 𝑌) = ( 𝑋) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
5 opoccl.b . . . . . 6 𝐵 = (Base‘𝐾)
6 opoccl.o . . . . . 6 = (oc‘𝐾)
75, 6opococ 38699 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
873adant3 1129 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
95, 6opococ 38699 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1093adant2 1128 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
118, 10eqeq12d 2744 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) = ( ‘( 𝑌)) ↔ 𝑋 = 𝑌))
124, 11imbitrid 243 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = ( 𝑋) → 𝑋 = 𝑌))
132, 12impbid2 225 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:  opcon2b  38701  omllaw4  38750  cmtbr2N  38757  cvrcmp2  38788  lhpmod2i2  39543  lhpmod6i1  39544
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