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Theorem cbvdisjdavw2 36232
Description: Change bound variable and domain in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvdisjdavw2.1 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvdisjdavw2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvdisjdavw2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑦𝐵 𝐷))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvdisjdavw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbvdisjdavw2.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
21eleq2d 2823 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐶𝑡𝐷))
3 cbvdisjdavw2.2 . . . 4 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
42, 3cbvrmodavw2 36226 . . 3 (𝜑 → (∃*𝑥𝐴 𝑡𝐶 ↔ ∃*𝑦𝐵 𝑡𝐷))
54albidv 1916 . 2 (𝜑 → (∀𝑡∃*𝑥𝐴 𝑡𝐶 ↔ ∀𝑡∃*𝑦𝐵 𝑡𝐷))
6 df-disj 5117 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑡∃*𝑥𝐴 𝑡𝐶)
7 df-disj 5117 . 2 (Disj 𝑦𝐵 𝐷 ↔ ∀𝑡∃*𝑦𝐵 𝑡𝐷)
85, 6, 73bitr4g 314 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑦𝐵 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1533   = wceq 1535  wcel 2104  ∃*wrmo 3375  Disj wdisj 5116
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-ex 1775  df-mo 2536  df-cleq 2725  df-clel 2812  df-rmo 3376  df-disj 5117
This theorem is referenced by: (None)
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