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Theorem cbvditgdavw2 36241
Description: Change bound variable and limits in a directed integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvditgdavw2.1 (𝜑𝐴 = 𝐵)
cbvditgdavw2.2 (𝜑𝐶 = 𝐷)
cbvditgdavw2.3 ((𝜑𝑥 = 𝑦) → 𝐸 = 𝐹)
Assertion
Ref Expression
cbvditgdavw2 (𝜑 → ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]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
StepHypRef Expression
1 cbvditgdavw2.1 . . . 4 (𝜑𝐴 = 𝐵)
2 cbvditgdavw2.2 . . . 4 (𝜑𝐶 = 𝐷)
31, 2breq12d 5162 . . 3 (𝜑 → (𝐴𝐶𝐵𝐷))
4 cbvditgdavw2.3 . . . 4 ((𝜑𝑥 = 𝑦) → 𝐸 = 𝐹)
51adantr 480 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
62adantr 480 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
75, 6oveq12d 7443 . . . 4 ((𝜑𝑥 = 𝑦) → (𝐴(,)𝐶) = (𝐵(,)𝐷))
84, 7cbvitgdavw2 36240 . . 3 (𝜑 → ∫(𝐴(,)𝐶)𝐸 d𝑥 = ∫(𝐵(,)𝐷)𝐹 d𝑦)
96, 5oveq12d 7443 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝐶(,)𝐴) = (𝐷(,)𝐵))
104, 9cbvitgdavw2 36240 . . . 4 (𝜑 → ∫(𝐶(,)𝐴)𝐸 d𝑥 = ∫(𝐷(,)𝐵)𝐹 d𝑦)
1110negeqd 11493 . . 3 (𝜑 → -∫(𝐶(,)𝐴)𝐸 d𝑥 = -∫(𝐷(,)𝐵)𝐹 d𝑦)
123, 8, 11ifbieq12d 4558 . 2 (𝜑 → if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥) = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦))
13 df-ditg 25878 . 2 ⨜[𝐴𝐶]𝐸 d𝑥 = if(𝐴𝐶, ∫(𝐴(,)𝐶)𝐸 d𝑥, -∫(𝐶(,)𝐴)𝐸 d𝑥)
14 df-ditg 25878 . 2 ⨜[𝐵𝐷]𝐹 d𝑦 = if(𝐵𝐷, ∫(𝐵(,)𝐷)𝐹 d𝑦, -∫(𝐷(,)𝐵)𝐹 d𝑦)
1512, 13, 143eqtr4g 2798 1 (𝜑 → ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑦)
Colors of variables: wff setvar class
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)
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