Theorem reueubd 3357

Description: Restricted existential uniqueness is equivalent to 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		df-reu 3070	. 2 ⊢ (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓))
2		reueubd.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)
3	2	ex 412	. . . 4 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝑉))
4	3	pm4.71rd 562	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝑥 ∈ 𝑉 ∧ 𝜓)))
5	4	eubidv 2586	. 2 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓)))
6	1, 5	bitr4id 289	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∃!weu 2568  ∃!wreu 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569  df-reu 3070

This theorem is referenced by:  frgreu  28533