Theorem reueubd 3438

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		reueubd.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)
2	1	ex 415	. . . 4 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝑉))
3	2	pm4.71rd 565	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝑥 ∈ 𝑉 ∧ 𝜓)))
4	3	eubidv 2672	. 2 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓)))
5		df-reu 3147	. 2 ⊢ (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓))
6	4, 5	syl6rbbr 292	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2114  ∃!weu 2653  ∃!wreu 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-mo 2622  df-eu 2654  df-reu 3147

This theorem is referenced by:  frgreu  28049