Theorem opcon3b 40069

Description: Contraposition law for orthoposets. (chcon3i 31947 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 6878	. . 3 ⊢ (𝑌 = 𝑋 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
2	1	eqcoms 2768	. 2 ⊢ (𝑋 = 𝑌 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))
3		fveq2 6878	. . . 4 ⊢ (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
4	3	eqcoms 2768	. . 3 ⊢ (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)))
5		opoccl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		opoccl.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
7	5, 6	opococ 40068	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
8	7	3adant3 1150	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
9	5, 6	opococ 40068	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	3adant2 1149	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
11	8, 10	eqeq12d 2776	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  Basecbs 17301  occoc 17350  OPcops 40045

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5665  df-iota 6489  df-fv 6541  df-ov 7416  df-oposet 40049

This theorem is used by:  opcon2b  40070  omllaw4  40119  cmtbr2N  40126  cvrcmp2  40157  lhpmod2i2  40911  lhpmod6i1  40912