Theorem eu5 1035

Description: Uniqueness in terms of "at most one".

Assertion

Ref	Expression
eu5	|- (E!xph <-> (E.xph /\ E*xph))

Proof of Theorem eu5

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 |- (ph -> A.yph)
2	1	eu3 1024	. 2 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
3	1	mo2 1026	. . 3 |- (E*xph <-> E.yA.x(ph -> x = y))
4	3	anbi2i 367	. 2 |- ((E.xph /\ E*xph) <-> (E.xph /\ E.yA.x(ph -> x = y)))
5	2, 4	bitr4 154	1 |- (E!xph <-> (E.xph /\ E*xph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007  E*wmo 1008

This theorem is referenced by:  eu4 1036  eumo 1037  exmoeu2 1040  euanv 1053  2euex 1061  2euswap 1065  2exeu 1066  2eu1 1067  reu5 1339  reuss 1577  aceq6b 3565  recmulpq 3864

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

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