Theorem itgeq2dv

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014)

	Ref	Expression
Hypothesis	itgeq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	itgeq2dv	$$⊢\phi \to {\int }_{A}Bdx={\int }_{A}Cdx$$

Proof

Step	Hyp	Ref	Expression
1		itgeq2dv.1	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
2	1	ralrimiva	$$⊢\phi \to \forall x\in AB=C$$
3		itgeq2	$$⊢\forall x\in AB=C\to {\int }_{A}Bdx={\int }_{A}Cdx$$
4	2 3	syl	$$⊢\phi \to {\int }_{A}Bdx={\int }_{A}Cdx$$