Theorem 2eu2 1493

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 1453	. . 3 |- (E!yE.xph -> E*yE.xph)
2		2moex 1483	. . 3 |- (E*yE.xph -> A.xE*yph)
3		2eu1 1492	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
4		pm3.26 326	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE.yph)
5	3, 4	syl6bi 221	. . 3 |- (A.xE*yph -> (E!xE!yph -> E!xE.yph))
6	1, 2, 5	3syl 20	. 2 |- (E!yE.xph -> (E!xE!yph -> E!xE.yph))
7		2exeu 1489	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
8	7	expcom 381	. 2 |- (E!yE.xph -> (E!xE.yph -> E!xE!yph))
9	6, 8	impbid 527	1 |- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995  E.wex 1021  E!weu 1422  E*wmo 1423

This theorem is referenced by:  2eu8 1499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425

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