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Theorem opcon3b 33501
 Description: Contraposition law for orthoposets. (chcon3i 27709 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 6103 . . 3 (𝑌 = 𝑋 → ( 𝑌) = ( 𝑋))
21eqcoms 2618 . 2 (𝑋 = 𝑌 → ( 𝑌) = ( 𝑋))
3 fveq2 6103 . . . 4 (( 𝑋) = ( 𝑌) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
43eqcoms 2618 . . 3 (( 𝑌) = ( 𝑋) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
5 opoccl.b . . . . . 6 𝐵 = (Base‘𝐾)
6 opoccl.o . . . . . 6 = (oc‘𝐾)
75, 6opococ 33500 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
873adant3 1074 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
95, 6opococ 33500 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1093adant2 1073 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
118, 10eqeq12d 2625 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) = ( ‘( 𝑌)) ↔ 𝑋 = 𝑌))
124, 11syl5ib 233 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = ( 𝑋) → 𝑋 = 𝑌))
132, 12impbid2 215 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Basecbs 15695  occoc 15776  OPcops 33477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oposet 33481 This theorem is referenced by:  opcon2b  33502  omllaw4  33551  cmtbr2N  33558  cvrcmp2  33589  lhpmod2i2  34342  lhpmod6i1  34343
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