Theorem cbvitgdavw

Description: Change bound variable in an integral. Deduction form. (Contributed by GG, 14-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		cbvitgdavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
2	1	fvoveq1d	$$⊢\left(\phi \wedge x=y\right)\to \Re \left(\frac{B}{{\mathrm{i}}^t}\right)=\Re \left(\frac{C}{{\mathrm{i}}^t}\right)$$
3		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
4	3	adantl	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in A\right)$$
5	4	anbi1d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(x\in A\wedge 0\le v\right)\leftrightarrow \left(y\in A\wedge 0\le v\right)\right)$$
6	5	ifbid	$$⊢\left(\phi \wedge x=y\right)\to if\left(\left(x\in A\wedge 0\le v\right),v,0\right)=if\left(\left(y\in A\wedge 0\le v\right),v,0\right)$$
7	2 6	csbeq12dv	$$⊢\left(\phi \wedge x=y\right)\to ⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)=⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)$$
8	7	cbvmptdavw	$$⊢\phi \to \left(x\in ℝ⟼⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)\right)=\left(y\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)\right)$$
9	8	fveq2d	$$⊢\phi \to {\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)\right)\right)={\int }^2\left(\left(y\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)\right)\right)$$
10	9	oveq2d	$$⊢\phi \to {\mathrm{i}}^t{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)\right)\right)={\mathrm{i}}^t{\int }^2\left(\left(y\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)\right)\right)$$
11	10	sumeq2sdv	$$⊢\phi \to \sum _{t=0}^3{\mathrm{i}}^t{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)\right)\right)=\sum _{t=0}^3{\mathrm{i}}^t{\int }^2\left(\left(y\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)\right)\right)$$
12		df-itg	$$⊢{\int }_{A}Bdx=\sum _{t=0}^3{\mathrm{i}}^t{\int }^2\left(\left(x\in ℝ⟼⦋\Re \left(\frac{B}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(x\in A\wedge 0\le v\right),v,0\right)\right)\right)$$
13		df-itg	$$⊢{\int }_{A}Cdy=\sum _{t=0}^3{\mathrm{i}}^t{\int }^2\left(\left(y\in ℝ⟼⦋\Re \left(\frac{C}{{\mathrm{i}}^t}\right)/v⦌if\left(\left(y\in A\wedge 0\le v\right),v,0\right)\right)\right)$$
14	11 12 13	3eqtr4g	$$⊢\phi \to {\int }_{A}Bdx={\int }_{A}Cdy$$