Theorem reu5 1339

Description: Restricted uniqueness in terms of "at most one".

Assertion

Ref	Expression
reu5	⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x(x ∈ A ∧ φ)))

Proof of Theorem reu5

Step	Hyp	Ref	Expression
1		eu5 1035	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃*x(x ∈ A ∧ φ)))
2		df-reu 1207	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
3		df-rex 1206	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4	3	anbi1i 368	. 2 ⊢ ((∃x ∈ A φ ∧ ∃*x(x ∈ A ∧ φ)) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃*x(x ∈ A ∧ φ)))
5	1, 2, 4	3bitr4 158	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x(x ∈ A ∧ φ)))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008   ∈ wcel 1092  ∃wrex 1202  ∃!wreu 1203

This theorem is referenced by:  supeu 2158  supsn 2168

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-rex 1206  df-reu 1207

metamath.org