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Theorem cbvreudavw2 36263
Description: Change bound variable and quantifier domain in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvreudavw2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvreudavw2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvreudavw2 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvreudavw2
StepHypRef Expression
1 simpr 484 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvreudavw2.2 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2834 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvreudavw2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 632 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65cbveudavw 36230 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑦(𝑦𝐵𝜒)))
7 df-reu 3380 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
8 df-reu 3380 . 2 (∃!𝑦𝐵 𝜒 ↔ ∃!𝑦(𝑦𝐵𝜒))
96, 7, 83bitr4g 314 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!weu 2567  ∃!wreu 3377
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-eu 2568  df-cleq 2728  df-clel 2815  df-reu 3380
This theorem is referenced by: (None)
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