Theorem ssdifd

Description: If A is contained in B , then ( A \ C ) is contained in ( B \ C ) . Deduction form of ssdif . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypothesis	ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
Assertion	ssdifd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssdif	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )