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Theorem wl-dfrexex 35034
 Description: Alternate definition of the restricted existential quantification (df-wl-rex 35030), according to the pattern given in df-wl-ral 35020. (Contributed by Wolf Lammen, 25-May-2023.)
Assertion
Ref Expression
wl-dfrexex (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfrexex
StepHypRef Expression
1 df-wl-ral 35020 . . 3 (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
21notbii 323 . 2 (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
3 df-wl-rex 35030 . 2 (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4 alinexa 1844 . . . . . 6 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
54imbi2i 339 . . . . 5 ((𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦𝐴 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
65albii 1821 . . . 4 (∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦𝐴 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
7 alinexa 1844 . . . 4 (∀𝑦(𝑦𝐴 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
86, 7bitri 278 . . 3 (∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ¬ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
98con2bii 361 . 2 (∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
102, 3, 93bitr4i 306 1 (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wl-ral 35015  ∃wl-rex 35016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-wl-ral 35020  df-wl-rex 35030 This theorem is referenced by: (None)
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