Theorem ssbrd

Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004)

	Ref	Expression
Hypothesis	ssbrd.1	$$⊢\phi \to A\subseteq B$$
Assertion	ssbrd	$$⊢\phi \to \left(CAD\to CBD\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssbrd.1	$$⊢\phi \to A\subseteq B$$
2	1	sseld	$$⊢\phi \to \left(⟨C,D⟩\in A\to ⟨C,D⟩\in B\right)$$
3		df-br	$$⊢CAD\leftrightarrow ⟨C,D⟩\in A$$
4		df-br	$$⊢CBD\leftrightarrow ⟨C,D⟩\in B$$
5	2 3 4	3imtr4g	$$⊢\phi \to \left(CAD\to CBD\right)$$