Theorem cbvditgvw2 36761

Description: Change bound variable and domain in a directed integral, using implicit substitution. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
cbvditgvw2.1	⊢ 𝐴 = 𝐵
cbvditgvw2.2	⊢ 𝐶 = 𝐷
cbvditgvw2.3	⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)

Assertion

Ref	Expression
cbvditgvw2	⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦

Distinct variable groups:   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 cbvditgvw2

Step	Hyp	Ref	Expression
1		cbvditgvw2.1	. . . 4 ⊢ 𝐴 = 𝐵
2		cbvditgvw2.2	. . . 4 ⊢ 𝐶 = 𝐷
3	1, 2	breq12i 5118	. . 3 ⊢ (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷)
4		cbvditgvw2.3	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
5	1, 2	oveq12i 7422	. . . . 5 ⊢ (𝐴(,)𝐶) = (𝐵(,)𝐷)
6	5	a1i 11	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴(,)𝐶) = (𝐵(,)𝐷))
7	4, 6	cbvitgvw2 36760	. . 3 ⊢ ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦
8	2	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
9	1	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
10	8, 9	oveq12d 7428	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶(,)𝐴) = (𝐷(,)𝐵))
11	4, 10	cbvitgvw2 36760	. . . 4 ⊢ ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦
12	11	negeqi 11445	. . 3 ⊢ -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦
13	3, 7, 12	ifbieq12i 4515	. 2 ⊢ if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
14		df-ditg 26006	. 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
15		df-ditg 26006	. 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
16	13, 14, 15	3eqtr4i 2796	1 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦

Syntax hints:   → wi 4   = wceq 1570  ifcif 4487   class class class wbr 5109  (class class class)co 7410   ≤ cle 11239  -cneg 11437  (,)cioo 13367  ∫citg 25777  ⨜cdit 26005

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-neg 11439  df-seq 14034  df-sum 15734  df-itg 25782  df-ditg 26006

This theorem is referenced by: (None)