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Theorem cbvreud 34185
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 1893 . . . 4 (𝜑 → Ⅎ𝑦 𝑥𝐴)
4 cbvreud.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
53, 4nfand 1879 . . 3 (𝜑 → Ⅎ𝑦(𝑥𝐴𝜓))
6 nfvd 1893 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 cbvreud.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
86, 7nfand 1879 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜒))
9 eleq1 2870 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109adantl 482 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
11 cbvreud.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
1211imp 407 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
1310, 12anbi12d 630 . . . 4 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐴𝜒)))
1413ex 413 . . 3 (𝜑 → (𝑥 = 𝑦 → ((𝑥𝐴𝜓) ↔ (𝑦𝐴𝜒))))
151, 2, 5, 8, 14cbveud 34184 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑦(𝑦𝐴𝜒)))
16 df-reu 3112 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
17 df-reu 3112 . 2 (∃!𝑦𝐴 𝜒 ↔ ∃!𝑦(𝑦𝐴𝜒))
1815, 16, 173bitr4g 315 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wnf 1765  wcel 2081  ∃!weu 2611  ∃!wreu 3107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-mo 2576  df-eu 2612  df-cleq 2788  df-clel 2863  df-reu 3112
This theorem is referenced by:  fvineqsneu  34223
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