Theorem resabs1d

Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	resabs1d.b	$$⊢\phi \to B\subseteq C$$
Assertion	resabs1d	$$⊢\phi \to {\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		resabs1d.b	$$⊢\phi \to B\subseteq C$$
2		resabs1	$$⊢B\subseteq C\to {\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_B$$
3	1 2	syl	$$⊢\phi \to {\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_B$$