Theorem ditgeq3dv

Description: Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014)

	Ref	Expression
Hypothesis	ditgeq3dv.1	$$⊢\left(\phi \wedge x\in ℝ\right)\to D=E$$
Assertion	ditgeq3dv	$$⊢\phi \to {\int }_A^{B}Ddx={\int }_A^{B}Edx$$

Proof

Step	Hyp	Ref	Expression
1		ditgeq3dv.1	$$⊢\left(\phi \wedge x\in ℝ\right)\to D=E$$
2	1	ralrimiva	$$⊢\phi \to \forall x\in ℝD=E$$
3		ditgeq3	$$⊢\forall x\in ℝD=E\to {\int }_A^{B}Ddx={\int }_A^{B}Edx$$
4	2 3	syl	$$⊢\phi \to {\int }_A^{B}Ddx={\int }_A^{B}Edx$$