Theorem chcon2i

Description: Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chcon2i	$$⊢A=\perp \left(B\right)\leftrightarrow B=\perp \left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chsscon2i	$$⊢A\subseteq \perp \left(B\right)\leftrightarrow B\subseteq \perp \left(A\right)$$
4	2 1	chsscon1i	$$⊢\perp \left(B\right)\subseteq A\leftrightarrow \perp \left(A\right)\subseteq B$$
5	3 4	anbi12i	$$⊢\left(A\subseteq \perp \left(B\right)\wedge \perp \left(B\right)\subseteq A\right)\leftrightarrow \left(B\subseteq \perp \left(A\right)\wedge \perp \left(A\right)\subseteq B\right)$$
6		eqss	$$⊢A=\perp \left(B\right)\leftrightarrow \left(A\subseteq \perp \left(B\right)\wedge \perp \left(B\right)\subseteq A\right)$$
7		eqss	$$⊢B=\perp \left(A\right)\leftrightarrow \left(B\subseteq \perp \left(A\right)\wedge \perp \left(A\right)\subseteq B\right)$$
8	5 6 7	3bitr4i	$$⊢A=\perp \left(B\right)\leftrightarrow B=\perp \left(A\right)$$