Theorem 2eu2 1068

Description: Double existential uniqueness.

Assertion

Ref	Expression
2eu2	|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 1037	. . 3 |- (E!yE.xph -> E*yE.xph)
2		2moex 1060	. . 3 |- (E*yE.xph -> A.xE*yph)
3		2eu1 1067	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
4		pm3.26 256	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE.yph)
5	3, 4	syl6bi 187	. . 3 |- (A.xE*yph -> (E!xE!yph -> E!xE.yph))
6	1, 2, 5	3syl 21	. 2 |- (E!yE.xph -> (E!xE!yph -> E!xE.yph))
7		2exeu 1066	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
8	7	exp 291	. . 3 |- (E!xE.yph -> (E!yE.xph -> E!xE!yph))
9	8	com12 13	. 2 |- (E!yE.xph -> (E!xE.yph -> E!xE!yph))
10	6, 9	impbid 397	1 |- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

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

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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