Description: Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ | ||
Assertion | chcon2i | ⊢ ( 𝐴 = ( ⊥ ‘ 𝐵 ) ↔ 𝐵 = ( ⊥ ‘ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
2 | chjcl.2 | ⊢ 𝐵 ∈ Cℋ | |
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 ) ) |