Theorem cbvditgdavw2 36526

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

Hypotheses

Ref	Expression
cbvditgdavw2.1	⊢ (𝜑 → 𝐴 = 𝐵)
cbvditgdavw2.2	⊢ (𝜑 → 𝐶 = 𝐷)
cbvditgdavw2.3	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐸 = 𝐹)

Assertion

Ref	Expression
cbvditgdavw2	⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷   𝑦,𝐸   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)   𝐸(𝑥)   𝐹(𝑦)

Proof of Theorem cbvditgdavw2

Step	Hyp	Ref	Expression
1		cbvditgdavw2.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2		cbvditgdavw2.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	breq12d 5085	. . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷))
4		cbvditgdavw2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐸 = 𝐹)
5	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
6	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
7	5, 6	oveq12d 7374	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐴(,)𝐶) = (𝐵(,)𝐷))
8	4, 7	cbvitgdavw2 36525	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦)
9	6, 5	oveq12d 7374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐶(,)𝐴) = (𝐷(,)𝐵))
10	4, 9	cbvitgdavw2 36525	. . . 4 ⊢ (𝜑 → ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦)
11	10	negeqd 11378	. . 3 ⊢ (𝜑 → -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦)
12	3, 8, 11	ifbieq12d 4483	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦))
13		df-ditg 25832	. 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
14		df-ditg 25832	. 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
15	12, 13, 14	3eqtr4g 2799	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ifcif 4454   class class class wbr 5072  (class class class)co 7356   ≤ cle 11171  -cneg 11369  (,)cioo 13289  ∫citg 25603  ⨜cdit 25831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-xp 5624  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-iota 6441  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-neg 11371  df-seq 13955  df-sum 15640  df-itg 25608  df-ditg 25832

This theorem is referenced by: (None)