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 )