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Theorem wl-dfreusb 34738
Description: An alternate definition of restricted existential uniqueness (df-wl-reu 34737) using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
Assertion
Ref Expression
wl-dfreusb (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfreusb
StepHypRef Expression
1 wl-dfrexsb 34732 . . 3 (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
2 wl-dfrmosb 34734 . . 3 (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
31, 2anbi12i 626 . 2 ((∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑) ↔ (∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
4 df-wl-reu 34737 . 2 (∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑))
5 df-eu 2647 . 2 (∃!𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)))
63, 4, 53bitr4i 304 1 (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771  [wsb 2060  wcel 2105  ∃*wmo 2613  ∃!weu 2646  wl-rex 34713  ∃*wl-rmo 34714  ∃!wl-reu 34715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-wl-ral 34717  df-wl-rex 34727  df-wl-rmo 34733  df-wl-reu 34737
This theorem is referenced by: (None)
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