Theorem cbvditgdavw2 36919

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 5116	. . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷))
4		cbvditgdavw2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐸 = 𝐹)
5	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
6	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
7	5, 6	oveq12d 7432	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐴(,)𝐶) = (𝐵(,)𝐷))
8	4, 7	cbvitgdavw2 36918	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦)
9	6, 5	oveq12d 7432	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐶(,)𝐴) = (𝐷(,)𝐵))
10	4, 9	cbvitgdavw2 36918	. . . 4 ⊢ (𝜑 → ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦)
11	10	negeqd 11476	. . 3 ⊢ (𝜑 → -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦)
12	3, 8, 11	ifbieq12d 4511	. 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 26075	. 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
14		df-ditg 26075	. 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
15	12, 13, 14	3eqtr4g 2820	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ifcif 4482   class class class wbr 5103  (class class class)co 7414   ≤ cle 11269  -cneg 11467  (,)cioo 13399  ∫citg 25847  ⨜cdit 26074

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-neg 11469  df-seq 14067  df-sum 15775  df-itg 25852  df-ditg 26075

This theorem is used by: (None)