Theorem ditgeq123dv

Description: Equality theorem for the directed integral. Deduction form. General version of ditgeq3sdv . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	ditgeq123dv.1	$$⊢\phi \to A=B$$
	ditgeq123dv.2	$$⊢\phi \to C=D$$
	ditgeq123dv.3	$$⊢\phi \to E=F$$
Assertion	ditgeq123dv	$$⊢\phi \to {\int }_A^{C}Edx={\int }_B^{D}Fdx$$

Proof

Step	Hyp	Ref	Expression
1		ditgeq123dv.1	$$⊢\phi \to A=B$$
2		ditgeq123dv.2	$$⊢\phi \to C=D$$
3		ditgeq123dv.3	$$⊢\phi \to E=F$$
4	1 2	breq12d	$$⊢\phi \to \left(A\le C\leftrightarrow B\le D\right)$$
5	1 2	oveq12d	$$⊢\phi \to \left(A,C\right)=\left(B,D\right)$$
6	5 3	itgeq12sdv	$$⊢\phi \to {\int }_{\left(A,C\right)}Edx={\int }_{\left(B,D\right)}Fdx$$
7	2 1	oveq12d	$$⊢\phi \to \left(C,A\right)=\left(D,B\right)$$
8	7 3	itgeq12sdv	$$⊢\phi \to {\int }_{\left(C,A\right)}Edx={\int }_{\left(D,B\right)}Fdx$$
9	8	negeqd	$$⊢\phi \to -{\int }_{\left(C,A\right)}Edx=-{\int }_{\left(D,B\right)}Fdx$$
10	4 6 9	ifbieq12d	$$⊢\phi \to if\left(A\le C,{\int }_{\left(A,C\right)}Edx,-{\int }_{\left(C,A\right)}Edx\right)=if\left(B\le D,{\int }_{\left(B,D\right)}Fdx,-{\int }_{\left(D,B\right)}Fdx\right)$$
11		df-ditg	$$⊢{\int }_A^{C}Edx=if\left(A\le C,{\int }_{\left(A,C\right)}Edx,-{\int }_{\left(C,A\right)}Edx\right)$$
12		df-ditg	$$⊢{\int }_B^{D}Fdx=if\left(B\le D,{\int }_{\left(B,D\right)}Fdx,-{\int }_{\left(D,B\right)}Fdx\right)$$
13	10 11 12	3eqtr4g	$$⊢\phi \to {\int }_A^{C}Edx={\int }_B^{D}Fdx$$