Metamath Proof Explorer


Theorem chcon2i

Description: Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1 𝐴C
chjcl.2 𝐵C
Assertion chcon2i ( 𝐴 = ( ⊥ ‘ 𝐵 ) ↔ 𝐵 = ( ⊥ ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ch0le.1 𝐴C
2 chjcl.2 𝐵C
3 1 2 chsscon2i ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) )
4 2 1 chsscon1i ( ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ↔ ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 )
5 3 4 anbi12i ( ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ∧ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ) ↔ ( 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ∧ ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ) )
6 eqss ( 𝐴 = ( ⊥ ‘ 𝐵 ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ∧ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ) )
7 eqss ( 𝐵 = ( ⊥ ‘ 𝐴 ) ↔ ( 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ∧ ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ) )
8 5 6 7 3bitr4i ( 𝐴 = ( ⊥ ‘ 𝐵 ) ↔ 𝐵 = ( ⊥ ‘ 𝐴 ) )