Theorem wl-dfreusb 34892

Description: An alternate definition of restricted existential uniqueness (df-wl-reu 34891) using substitution. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

Ref	Expression
wl-dfreusb	⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfreusb

Step	Hyp	Ref	Expression
1		wl-dfrexsb 34886	. . 3 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
2		wl-dfrmosb 34888	. . 3 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
3	1, 2	anbi12i 628	. 2 ⊢ ((∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑) ↔ (∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
4		df-wl-reu 34891	. 2 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑))
5		df-eu 2653	. 2 ⊢ (∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6	3, 4, 5	3bitr4i 305	1 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1779  [wsb 2068   ∈ 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-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-wl-ral 34871  df-wl-rex 34881  df-wl-rmo 34887  df-wl-reu 34891

This theorem is referenced by: (None)