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Theorem cbvdisjdavw 36247
Description: Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvdisjdavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvdisjdavw (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvdisjdavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbvdisjdavw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
21eleq2d 2826 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐵𝑡𝐶))
32cbvrmodavw 36231 . . 3 (𝜑 → (∃*𝑥𝐴 𝑡𝐵 ↔ ∃*𝑦𝐴 𝑡𝐶))
43albidv 1920 . 2 (𝜑 → (∀𝑡∃*𝑥𝐴 𝑡𝐵 ↔ ∀𝑡∃*𝑦𝐴 𝑡𝐶))
5 df-disj 5109 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑡∃*𝑥𝐴 𝑡𝐵)
6 df-disj 5109 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑡∃*𝑦𝐴 𝑡𝐶)
74, 5, 63bitr4g 314 1 (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  ∃*wrmo 3378  Disj wdisj 5108
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 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-cleq 2728  df-clel 2815  df-rmo 3379  df-disj 5109
This theorem is referenced by: (None)
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