Theorem cbvditgdavw2 36277

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 5154	. . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷))
4		cbvditgdavw2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐸 = 𝐹)
5	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
6	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
7	5, 6	oveq12d 7447	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐴(,)𝐶) = (𝐵(,)𝐷))
8	4, 7	cbvitgdavw2 36276	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦)
9	6, 5	oveq12d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐶(,)𝐴) = (𝐷(,)𝐵))
10	4, 9	cbvitgdavw2 36276	. . . 4 ⊢ (𝜑 → ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦)
11	10	negeqd 11498	. . 3 ⊢ (𝜑 → -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦)
12	3, 8, 11	ifbieq12d 4552	. 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 25872	. 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
14		df-ditg 25872	. 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
15	12, 13, 14	3eqtr4g 2801	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ifcif 4524   class class class wbr 5141  (class class class)co 7429   ≤ cle 11292  -cneg 11489  (,)cioo 13383  ∫citg 25643  ⨜cdit 25871

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-neg 11491  df-seq 14039  df-sum 15719  df-itg 25648  df-ditg 25872

This theorem is referenced by: (None)