Theorem reueubd 3164

Description: Restricted existential uniqueness is equivalent with existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)

Hypothesis

Ref	Expression
reueubd.1	⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)

Assertion

Ref	Expression
reueubd	⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem reueubd

Step	Hyp	Ref	Expression
1		reueubd.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)
2	1	ex 450	. . . 4 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝑉))
3	2	pm4.71rd 667	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝑥 ∈ 𝑉 ∧ 𝜓)))
4	3	eubidv 2490	. 2 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓)))
5		df-reu 2919	. 2 ⊢ (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓))
6	4, 5	syl6rbbr 279	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  ∃!weu 2470  ∃!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-reu 2919

This theorem is referenced by:  frgreu  27132