Theorem opcon3b

Description: Contraposition law for orthoposets. ( chcon3i analog.) (Contributed by NM, 8-Nov-2011)

	Ref	Expression
Hypotheses	opoccl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opoccl.o	⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion	opcon3b	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 ↔ ( ⊥ ‘ 𝑌 ) = ( ⊥ ‘ 𝑋 ) ) )

Proof

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