Theorem cbvitgdavw 36255

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

Hypothesis

Ref	Expression
cbvitgdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvitgdavw	⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvitgdavw

Dummy variables 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvitgdavw.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
2	1	fvoveq1d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) = (ℜ‘(𝐶 / (i↑𝑡))))
3		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	4	anbi1d 631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣)))
6	5	ifbid 4500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
7	2, 6	csbeq12dv 3860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	7	cbvmptdavw 36241	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))
9	8	fveq2d 6826	. . . 4 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
10	9	oveq2d 7365	. . 3 ⊢ (𝜑 → ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
11	10	sumeq2sdv 15610	. 2 ⊢ (𝜑 → Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
12		df-itg 25522	. 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
13		df-itg 25522	. 2 ⊢ ∫𝐴𝐶 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
14	11, 12, 13	3eqtr4g 2789	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⦋csb 3851  ifcif 4476   class class class wbr 5092   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  ℝcr 11008  0cc0 11009  ici 11011   · cmul 11014   ≤ cle 11150   / cdiv 11777  3c3 12184  ...cfz 13410  ↑cexp 13968  ℜcre 15004  Σcsu 15593  ∫₂citg2 25515  ∫citg 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-xp 5625  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-iota 6438  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-seq 13909  df-sum 15594  df-itg 25522

This theorem is referenced by:  cbvditgdavw  36256