Theorem wl-dfreuv 34893

Description: Alternate definition of restricted existential uniqueness (df-wl-reu 34891) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

Ref	Expression
wl-dfreuv	⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfreuv

Step	Hyp	Ref	Expression
1		wl-dfrexv 34884	. . 3 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		wl-dfrmov 34889	. . 3 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3	1, 2	anbi12i 628	. 2 ⊢ ((∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-wl-reu 34891	. 2 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑))
5		df-eu 2653	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6	3, 4, 5	3bitr4i 305	1 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1779   ∈ wcel 2113  ∃*wmo 2619  ∃!weu 2652  ∃wl-rex 34867  ∃*wl-rmo 34868  ∃!wl-reu 34869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-11 2160

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-mo 2621  df-eu 2653  df-clel 2892  df-wl-ral 34871  df-wl-rex 34881  df-wl-rmo 34887  df-wl-reu 34891

This theorem is referenced by: (None)