Theorem chcon1i

Description: Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chcon1i	$$⊢\perp \left(A\right)=B\leftrightarrow \perp \left(B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	2 1	chcon2i	$$⊢B=\perp \left(A\right)\leftrightarrow A=\perp \left(B\right)$$
4		eqcom	$$⊢\perp \left(A\right)=B\leftrightarrow B=\perp \left(A\right)$$
5		eqcom	$$⊢\perp \left(B\right)=A\leftrightarrow A=\perp \left(B\right)$$
6	3 4 5	3bitr4i	$$⊢\perp \left(A\right)=B\leftrightarrow \perp \left(B\right)=A$$