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Theorem opcon3b 37661
Description: Contraposition law for orthoposets. (chcon3i 30411 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 6843 . . 3 (𝑌 = 𝑋 → ( 𝑌) = ( 𝑋))
21eqcoms 2745 . 2 (𝑋 = 𝑌 → ( 𝑌) = ( 𝑋))
3 fveq2 6843 . . . 4 (( 𝑋) = ( 𝑌) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
43eqcoms 2745 . . 3 (( 𝑌) = ( 𝑋) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
5 opoccl.b . . . . . 6 𝐵 = (Base‘𝐾)
6 opoccl.o . . . . . 6 = (oc‘𝐾)
75, 6opococ 37660 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
873adant3 1133 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
95, 6opococ 37660 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1093adant2 1132 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
118, 10eqeq12d 2753 . . 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 1088   = wceq 1542  wcel 2107  cfv 6497  Basecbs 17084  occoc 17142  OPcops 37637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oposet 37641
This theorem is referenced by:  opcon2b  37662  omllaw4  37711  cmtbr2N  37718  cvrcmp2  37749  lhpmod2i2  38504  lhpmod6i1  38505
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