Theorem opcon1b 33503

Description: Orthocomplement contraposition law. (negcon1 10212 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 33502	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋)))
4		eqcom 2617	. . 3 ⊢ (( ⊥ ‘𝑌) = 𝑋 ↔ 𝑋 = ( ⊥ ‘𝑌))
5		eqcom 2617	. . 3 ⊢ (( ⊥ ‘𝑋) = 𝑌 ↔ 𝑌 = ( ⊥ ‘𝑋))
6	3, 4, 5	3bitr4g 302	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = 𝑌))
7	6	bicomd 212	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Basecbs 15695  occoc 15776  OPcops 33477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oposet 33481

This theorem is referenced by:  opoc0  33508