Theorem opcon1b 37458

Description: Orthocomplement contraposition law. (negcon1 11366 analog.) (Contributed by NM, 24-Jan-2012.)

Hypotheses

Ref	Expression
opoccl.b	⊢ 𝐵 = (Base‘𝐾)
opoccl.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
opcon1b	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))

Proof of Theorem opcon1b

Step	Hyp	Ref	Expression
1		opoccl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		opoccl.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
3	1, 2	opcon2b 37457	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋)))
4		eqcom 2743	. . 3 ⊢ (( ⊥ ‘𝑌) = 𝑋 ↔ 𝑋 = ( ⊥ ‘𝑌))
5		eqcom 2743	. . 3 ⊢ (( ⊥ ‘𝑋) = 𝑌 ↔ 𝑌 = ( ⊥ ‘𝑋))
6	3, 4, 5	3bitr4g 313	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = 𝑌))
7	6	bicomd 222	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ‘cfv 6473  Basecbs 17001  occoc 17059  OPcops 37432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5247

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-dm 5624  df-iota 6425  df-fv 6481  df-ov 7332  df-oposet 37436

This theorem is referenced by:  opoc0  37463