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Theorem cbvreud 34809
 Description: Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.)
Hypotheses
Ref Expression
cbvreud.1 𝑥𝜑
cbvreud.2 𝑦𝜑
cbvreud.3 (𝜑 → Ⅎ𝑦𝜓)
cbvreud.4 (𝜑 → Ⅎ𝑥𝜒)
cbvreud.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvreud (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbvreud
StepHypRef Expression
1 cbvreud.1 . . 3 𝑥𝜑
2 cbvreud.2 . . 3 𝑦𝜑
3 nfvd 1916 . . . 4 (𝜑 → Ⅎ𝑦 𝑥𝐴)
4 cbvreud.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
53, 4nfand 1898 . . 3 (𝜑 → Ⅎ𝑦(𝑥𝐴𝜓))
6 nfvd 1916 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 cbvreud.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
86, 7nfand 1898 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜒))
9 eleq1 2877 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109adantl 485 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
11 cbvreud.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
1211imp 410 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
1310, 12anbi12d 633 . . . 4 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐴𝜒)))
1413ex 416 . . 3 (𝜑 → (𝑥 = 𝑦 → ((𝑥𝐴𝜓) ↔ (𝑦𝐴𝜒))))
151, 2, 5, 8, 14cbveud 34808 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑦(𝑦𝐴𝜒)))
16 df-reu 3113 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
17 df-reu 3113 . 2 (∃!𝑦𝐴 𝜒 ↔ ∃!𝑦(𝑦𝐴𝜒))
1815, 16, 173bitr4g 317 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∃!weu 2628  ∃!wreu 3108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629  df-cleq 2791  df-clel 2870  df-reu 3113 This theorem is referenced by:  fvineqsneu  34847
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