Description: Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ | ||
Assertion | chcon1i | ⊢ ( ( ⊥ ‘ 𝐴 ) = 𝐵 ↔ ( ⊥ ‘ 𝐵 ) = 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
2 | chjcl.2 | ⊢ 𝐵 ∈ Cℋ | |
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 ) |