Theorem cbvitgdavw 36793

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 7432	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) = (ℜ‘(𝐶 / (i↑𝑡))))
3		eleq1w 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	4	anbi1d 642	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣)))
6	5	ifbid 4511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
7	2, 6	csbeq12dv 3862	. . . . . 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 36779	. . . . 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 6885	. . . 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 7426	. . 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 15750	. 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 25782	. 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 25782	. 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 2823	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⦋csb 3853  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  ℝcr 11094  0cc0 11095  ici 11097   · cmul 11100   ≤ cle 11239   / cdiv 11866  3c3 12291  ...cfz 13530  ↑cexp 14093  ℜcre 15144  Σcsu 15733  ∫₂citg2 25775  ∫citg 25777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seq 14034  df-sum 15734  df-itg 25782

This theorem is referenced by:  cbvditgdavw  36794