Theorem wl-dfreuf 34895

Description: Restricted existential uniqueness (df-wl-reu 34892) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

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

Proof of Theorem wl-dfreuf

Step	Hyp	Ref	Expression
1		wl-dfrexf 34883	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
2		wl-dfrmof 34891	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	anbi12d 632	. 2 ⊢ (Ⅎ𝑥𝐴 → ((∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
4		df-wl-reu 34892	. 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	3bitr4g 316	1 ⊢ (Ⅎ𝑥𝐴 → (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779   ∈ wcel 2113  ∃*wmo 2619  ∃!weu 2652  Ⅎwnfc 2960  ∃wl-rex 34868  ∃*wl-rmo 34869  ∃!wl-reu 34870

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-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-mo 2621  df-eu 2653  df-clel 2892  df-nfc 2962  df-wl-ral 34872  df-wl-rex 34882  df-wl-rmo 34888  df-wl-reu 34892

This theorem is referenced by: (None)