Theorem reu5 1925

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 1407	. 2 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
2		df-reu 1648	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
3		df-rex 1647	. . 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 956  E.wex 978  E!weu 1378  E*wmo 1379  E.wrex 1643  E!wreu 1644

This theorem is referenced by:  reu4 1930  mouniss 2885  fncnv 3553  supeu 4558  suppr 4570  supsnALT 4572  spweu 8599  cnlnadjeu 9948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-rex 1647  df-reu 1648

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