Theorem opcon3b 39855

Description: Contraposition law for orthoposets. (chcon3i 31755 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 6879	. . 3 ⊢ (𝑌 = 𝑋 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
2	1	eqcoms 2777	. 2 ⊢ (𝑋 = 𝑌 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
3		fveq2 6879	. . . 4 ⊢ (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
4	3	eqcoms 2777	. . 3 ⊢ (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
5		opoccl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		opoccl.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
7	5, 6	opococ 39854	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
8	7	3adant3 1148	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
9	5, 6	opococ 39854	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	3adant2 1147	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
11	8, 10	eqeq12d 2785	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑋 = 𝑌))
12	4, 11	imbitrid 247	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → 𝑋 = 𝑌))
13	2, 12	impbid2 229	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ‘cfv 6533  Basecbs 17265  occoc 17314  OPcops 39831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-dm 5669  df-iota 6489  df-fv 6541  df-ov 7411  df-oposet 39835

This theorem is referenced by:  opcon2b  39856  omllaw4  39905  cmtbr2N  39912  cvrcmp2  39943  lhpmod2i2  40697  lhpmod6i1  40698