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Metamath Proof Explorer

Theorem chcon1i

Description: Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1 ⊢ A ∈ C ℋ
chjcl.2 ⊢ B ∈ C ℋ
Assertion chcon1i ⊢ ⊥ ⁡ A = B ↔ ⊥ ⁡ B = A

Proof

Step Hyp Ref Expression
1 ch0le.1 ⊢ A ∈ C ℋ
2 chjcl.2 ⊢ B ∈ C ℋ
3 2 1 chcon2i ⊢ B = ⊥ ⁡ A ↔ A = ⊥ ⁡ B
4 eqcom ⊢ ⊥ ⁡ A = B ↔ B = ⊥ ⁡ A
5 eqcom ⊢ ⊥ ⁡ B = A ↔ A = ⊥ ⁡ B
6 3 4 5 3bitr4i ⊢ ⊥ ⁡ A = B ↔ ⊥ ⁡ B = A