Theorem reu5 1929

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 1409	. 2 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
2		df-reu 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
3		df-rex 1650	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4	3	anbi1i 481	. 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 183	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E*x(x e. A /\ ph)))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958  E.wex 980  E!weu 1380  E*wmo 1381  E.wrex 1646  E!wreu 1647

This theorem is referenced by:  reu4 1934  mouniss 2890  fncnv 3561  supeu 4578  suppr 4590  supsnALT 4592  spweu 8657  cnlnadjeu 10010

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-rex 1650  df-reu 1651

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