Theorem 2euexOLD 1845

Description: Double quantification with existential uniqueness.

Assertion

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

Proof of Theorem 2euexOLD

Step	Hyp	Ref	Expression
1		eu5 1805	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2		hbe1 1363	. . . . . . 7 |- (E.yph -> A.yE.yph)
3	2	hbmo 1803	. . . . . 6 |- (E*xE.yph -> A.yE*xE.yph)
4	3	19.41 1448	. . . . 5 |- (E.y(E.xph /\ E*xE.yph) <-> (E.yE.xph /\ E*xE.yph))
5	4	biimpri 169	. . . 4 |- ((E.yE.xph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
6		excom 1393	. . . 4 |- (E.xE.yph <-> E.yE.xph)
7	5, 6	sylanb 498	. . 3 |- ((E.xE.yph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
8		2moex 1843	. . . . . . 7 |- (E*xE.yph -> A.yE*xph)
9	8	19.21bi 1408	. . . . . 6 |- (E*xE.yph -> E*xph)
10	9	anim2i 362	. . . . 5 |- ((E.xph /\ E*xE.yph) -> (E.xph /\ E*xph))
11		eu5 1805	. . . . 5 |- (E!xph <-> (E.xph /\ E*xph))
12	10, 11	sylibr 217	. . . 4 |- ((E.xph /\ E*xE.yph) -> E!xph)
13	12	eximi 1387	. . 3 |- (E.y(E.xph /\ E*xE.yph) -> E.yE!xph)
14	7, 13	syl 12	. 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
15	1, 14	sylbi 216	1 |- (E!xE.yph -> E.yE!xph)

Syntax hints:   -> wi 3   /\ wa 240  E.wex 1326  E!weu 1771  E*wmo 1772

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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