Theorem 2eu7 1495

Description: Two equivalent expressions for double existential uniqueness.

Assertion

Ref	Expression
2eu7	|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		hbe1 1052	. . . 4 |- (E.xph -> A.xE.xph)
2	1	hbeu 1428	. . 3 |- (E!yE.xph -> A.xE!yE.xph)
3	2	euan 1467	. 2 |- (E!x(E!yE.xph /\ E.yph) <-> (E!yE.xph /\ E!xE.yph))
4		ancom 437	. . . . 5 |- ((E.xph /\ E.yph) <-> (E.yph /\ E.xph))
5	4	eubii 1426	. . . 4 |- (E!y(E.xph /\ E.yph) <-> E!y(E.yph /\ E.xph))
6		hbe1 1052	. . . . 5 |- (E.yph -> A.yE.yph)
7	6	euan 1467	. . . 4 |- (E!y(E.yph /\ E.xph) <-> (E.yph /\ E!yE.xph))
8		ancom 437	. . . 4 |- ((E.yph /\ E!yE.xph) <-> (E!yE.xph /\ E.yph))
9	5, 7, 8	3bitri 175	. . 3 |- (E!y(E.xph /\ E.yph) <-> (E!yE.xph /\ E.yph))
10	9	eubii 1426	. 2 |- (E!xE!y(E.xph /\ E.yph) <-> E!x(E!yE.xph /\ E.yph))
11		ancom 437	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E!yE.xph /\ E!xE.yph))
12	3, 10, 11	3bitr4ri 182	1 |- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Syntax hints:   <-> wb 144   /\ wa 221  E.wex 1016  E!weu 1419

This theorem is referenced by:  2eu8 1496

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422

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