Theorem wl-dfreuv 34873

Description: Alternate definition of restricted existential uniqueness (df-wl-reu 34871) 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 34864	. . 3 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		wl-dfrmov 34869	. . 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 34871	. 2 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑))
5		df-eu 2654	. 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 1780   ∈ wcel 2114  ∃*wmo 2620  ∃!weu 2653  ∃wl-rex 34847  ∃*wl-rmo 34848  ∃!wl-reu 34849

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  ax-6 1970  ax-7 2015  ax-8 2116  ax-11 2161

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-mo 2622  df-eu 2654  df-clel 2893  df-wl-ral 34851  df-wl-rex 34861  df-wl-rmo 34867  df-wl-reu 34871

This theorem is referenced by: (None)