Theorem cbvditgvw2 36869

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 5112	. . 3 ⊢ (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐷)
4		cbvditgvw2.3	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)
5	1, 2	oveq12i 7425	. . . . 5 ⊢ (𝐴(,)𝐶) = (𝐵(,)𝐷)
6	5	a1i 11	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴(,)𝐶) = (𝐵(,)𝐷))
7	4, 6	cbvitgvw2 36868	. . 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 7431	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶(,)𝐴) = (𝐷(,)𝐵))
11	4, 10	cbvitgvw2 36868	. . . 4 ⊢ ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦
12	11	negeqi 11474	. . 3 ⊢ -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦
13	3, 7, 12	ifbieq12i 4510	. 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 26074	. 2 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = if(𝐴 ≤ 𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
15		df-ditg 26074	. 2 ⊢ ⨜[𝐵 → 𝐷]𝐹 d𝑦 = if(𝐵 ≤ 𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
16	13, 14, 15	3eqtr4i 2793	1 ⊢ ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ifcif 4482   class class class wbr 5103  (class class class)co 7413   ≤ cle 11268  -cneg 11466  (,)cioo 13398  ∫citg 25846  ⨜cdit 26073

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 7416  df-oprab 7417  df-mpo 7418  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-neg 11468  df-seq 14066  df-sum 15774  df-itg 25851  df-ditg 26074

This theorem is used by: (None)