Theorem cbvditgdavw 36256

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

Hypothesis

Ref	Expression
cbvditgdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)

Assertion

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

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

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvditgdavw

Step	Hyp	Ref	Expression
1		cbvditgdavw.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
2	1	cbvitgdavw 36255	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐷 d𝑦)
3	1	cbvitgdavw 36255	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐷 d𝑦)
4	3	negeqd 11357	. . 3 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐷 d𝑦)
5	2, 4	ifeq12d 4498	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐷 d𝑦, -∫(𝐵(,)𝐴)𝐷 d𝑦))
6		df-ditg 25746	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
7		df-ditg 25746	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐷 d𝑦 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐷 d𝑦, -∫(𝐵(,)𝐴)𝐷 d𝑦)
8	5, 6, 7	3eqtr4g 2789	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑦)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ifcif 4476   class class class wbr 5092  (class class class)co 7349   ≤ cle 11150  -cneg 11348  (,)cioo 13248  ∫citg 25517  ⨜cdit 25745

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-xp 5625  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-iota 6438  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-neg 11350  df-seq 13909  df-sum 15594  df-itg 25522  df-ditg 25746

This theorem is referenced by: (None)