Theorem reu5 1339

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

Assertion

Ref	Expression
reu5	|- (E!x e. A ph <-> (E.x e. A ph /\ E*x(x e. A /\ ph)))

Proof of Theorem reu5

Step	Hyp	Ref	Expression
1		eu5 1035	. 2 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
2		df-reu 1207	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
3		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4	3	anbi1i 368	. 2 |- ((E.x e. A ph /\ E*x(x e. A /\ ph)) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
5	1, 2, 4	3bitr4 158	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E*x(x e. A /\ ph)))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678  E!weu 1007  E*wmo 1008   e. wcel 1092  E.wrex 1202  E!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

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