Theorem chcon3i

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	chcon3i	⊢ ( 𝐴 = 𝐵 ↔ ( ⊥ ‘ 𝐵 ) = ( ⊥ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chsscon3i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) )
4	2 1	chsscon3i	⊢ ( 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 ) )