Theorem 2eu2 1450

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 1411	. . 3 |- (E!yE.xph -> E*yE.xph)
2		2moex 1440	. . 3 |- (E*yE.xph -> A.xE*yph)
3		2eu1 1449	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
4		pm3.26 319	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE.yph)
5	3, 4	syl6bi 214	. . 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 1446	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
8	7	expcom 374	. 2 |- (E!yE.xph -> (E!xE.yph -> E!xE!yph))
9	6, 8	impbid 516	1 |- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954  E.wex 980  E!weu 1380  E*wmo 1381

This theorem is referenced by:  2eu8 1456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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