Metamath Proof Explorer

Theorem chcon3i

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

Ref Expression
Hypotheses ch0le.1 ⊢ A ∈ C ℋ
chjcl.2 ⊢ B ∈ C ℋ
Assertion chcon3i ⊢ A = B ↔ ⊥ ⁡ B = ⊥ ⁡ A

Proof

Step Hyp Ref Expression
1 ch0le.1 ⊢ A ∈ C ℋ
2 chjcl.2 ⊢ B ∈ C ℋ
3 1 2 chsscon3i ⊢ A ⊆ B ↔ ⊥ ⁡ B ⊆ ⊥ ⁡ A
4 2 1 chsscon3i ⊢ 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