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 𝐴C
chjcl.2 𝐵C
Assertion chcon1i ( ( ⊥ ‘ 𝐴 ) = 𝐵 ↔ ( ⊥ ‘ 𝐵 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 ch0le.1 𝐴C
2 chjcl.2 𝐵C
3 2 1 chcon2i ( 𝐵 = ( ⊥ ‘ 𝐴 ) ↔ 𝐴 = ( ⊥ ‘ 𝐵 ) )
4 eqcom ( ( ⊥ ‘ 𝐴 ) = 𝐵𝐵 = ( ⊥ ‘ 𝐴 ) )
5 eqcom ( ( ⊥ ‘ 𝐵 ) = 𝐴𝐴 = ( ⊥ ‘ 𝐵 ) )
6 3 4 5 3bitr4i ( ( ⊥ ‘ 𝐴 ) = 𝐵 ↔ ( ⊥ ‘ 𝐵 ) = 𝐴 )