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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 ∈ Cℋ
chjcl.2 ⊢ 𝐵 ∈ Cℋ
Assertion chcon1i ( ( ⊥ ‘ 𝐴 ) = 𝐵 ↔ ( ⊥ ‘ 𝐵 ) = 𝐴 )

Proof

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 )