Theorem opcon3b 37196

Description: Contraposition law for orthoposets. (chcon3i 29814 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))

Proof of Theorem opcon3b

Step	Hyp	Ref	Expression
1		fveq2 6767	. . 3 ⊢ (𝑌 = 𝑋 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
2	1	eqcoms 2746	. 2 ⊢ (𝑋 = 𝑌 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
3		fveq2 6767	. . . 4 ⊢ (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
4	3	eqcoms 2746	. . 3 ⊢ (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
5		opoccl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		opoccl.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
7	5, 6	opococ 37195	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
8	7	3adant3 1131	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
9	5, 6	opococ 37195	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	3adant2 1130	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
11	8, 10	eqeq12d 2754	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑋 = 𝑌))
12	4, 11	syl5ib 243	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → 𝑋 = 𝑌))
13	2, 12	impbid2 225	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ‘cfv 6427  Basecbs 16900  occoc 16958  OPcops 37172

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5229

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-dm 5595  df-iota 6385  df-fv 6435  df-ov 7271  df-oposet 37176

This theorem is referenced by:  opcon2b  37197  omllaw4  37246  cmtbr2N  37253  cvrcmp2  37284  lhpmod2i2  38038  lhpmod6i1  38039