Theorem cbvitgdavw 36260

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 7451	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) = (ℜ‘(𝐶 / (i↑𝑡))))
3		eleq1w 2823	. . . . . . . . . 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 4547	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
7	2, 6	csbeq12dv 3907	. . . . . 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 36246	. . . . 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 6908	. . . 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 7445	. . 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 15735	. 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 25648	. 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 25648	. 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 2801	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⦋csb 3898  ifcif 4524   class class class wbr 5141   ↦ cmpt 5223  ‘cfv 6559  (class class class)co 7429  ℝcr 11150  0cc0 11151  ici 11153   · cmul 11156   ≤ cle 11292   / cdiv 11916  3c3 12318  ...cfz 13543  ↑cexp 14098  ℜcre 15132  Σcsu 15718  ∫₂citg2 25641  ∫citg 25643

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-seq 14039  df-sum 15719  df-itg 25648

This theorem is referenced by:  cbvditgdavw  36261