Theorem 2eu2 1854

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 1807	. . 3 |- (E!yE.xph -> E*yE.xph)
2		2moex 1843	. . 3 |- (E*yE.xph -> A.xE*yph)
3		2eu1 1853	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
4		simpl 346	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE.yph)
5	3, 4	syl6bi 231	. . 3 |- (A.xE*yph -> (E!xE!yph -> E!xE.yph))
6	1, 2, 5	3syl 24	. 2 |- (E!yE.xph -> (E!xE!yph -> E!xE.yph))
7		2exeu 1850	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
8	7	expcom 403	. 2 |- (E!yE.xph -> (E!xE.yph -> E!xE!yph))
9	6, 8	impbid 574	1 |- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem is referenced by:  2eu8 1860

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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