Theorem 2euex 1481

Description: Double quantification with existential uniqueness.

Assertion

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

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 1448	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2		hbe1 1052	. . . . . . 7 |- (E.yph -> A.yE.yph)
3	2	hbmo 1446	. . . . . 6 |- (E*xE.yph -> A.yE*xE.yph)
4	3	19.41 1131	. . . . 5 |- (E.y(E.xph /\ E*xE.yph) <-> (E.yE.xph /\ E*xE.yph))
5	4	biimpri 150	. . . 4 |- ((E.yE.xph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
6		excom 1082	. . . 4 |- (E.xE.yph <-> E.yE.xph)
7	5, 6	sylanb 451	. . 3 |- ((E.xE.yph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
8		2moex 1480	. . . . . . 7 |- (E*xE.yph -> A.yE*xph)
9	8	19.21bi 1096	. . . . . 6 |- (E*xE.yph -> E*xph)
10	9	anim2i 333	. . . . 5 |- ((E.xph /\ E*xE.yph) -> (E.xph /\ E*xph))
11		eu5 1448	. . . . 5 |- (E!xph <-> (E.xph /\ E*xph))
12	10, 11	sylibr 198	. . . 4 |- ((E.xph /\ E*xE.yph) -> E!xph)
13	12	19.22i 1076	. . 3 |- (E.y(E.xph /\ E*xE.yph) -> E.yE!xph)
14	7, 13	syl 10	. 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
15	1, 14	sylbi 197	1 |- (E!xE.yph -> E.yE!xph)

Syntax hints:   -> wi 3   /\ wa 221  E.wex 1016  E!weu 1419  E*wmo 1420

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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