Theorem cbvditgdavw 36225

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 36224	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐷 d𝑦)
3	1	cbvitgdavw 36224	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐷 d𝑦)
4	3	negeqd 11493	. . 3 ⊢ (𝜑 → -∫(𝐵(,)𝐴)𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐷 d𝑦)
5	2, 4	ifeq12d 4551	. 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 25878	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
7		df-ditg 25878	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐷 d𝑦 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐷 d𝑦, -∫(𝐵(,)𝐴)𝐷 d𝑦)
8	5, 6, 7	3eqtr4g 2798	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑦)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1535  ifcif 4530   class class class wbr 5149  (class class class)co 7425   ≤ cle 11287  -cneg 11484  (,)cioo 13377  ∫citg 25648  ⨜cdit 25877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-neg 11486  df-seq 14029  df-sum 15709  df-itg 25653  df-ditg 25878

This theorem is referenced by: (None)