Metamath Proof Explorer


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 ∧ 𝑋𝐵𝑌𝐵 ) → ( 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 ∧ 𝑋𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
8 7 3adant3 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
9 1 2 opococ ( ( 𝐾 ∈ OP ∧ 𝑌𝐵 ) → ( ‘ ( 𝑌 ) ) = 𝑌 )
10 9 3adant2 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( 𝑌 ) ) = 𝑌 )
11 8 10 eqeq12d ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( ‘ ( 𝑋 ) ) = ( ‘ ( 𝑌 ) ) ↔ 𝑋 = 𝑌 ) )
12 6 11 syl5ib ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑌 ) = ( 𝑋 ) → 𝑋 = 𝑌 ) )
13 4 12 impbid2 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 = 𝑌 ↔ ( 𝑌 ) = ( 𝑋 ) ) )