Theorem 2euex 1061

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 1035	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2		hbe1 709	. . . . . . 7 |- (E.yph -> A.yE.yph)
3	2	hbmo 1033	. . . . . 6 |- (E*xE.yph -> A.yE*xE.yph)
4	3	19.41 774	. . . . 5 |- (E.y(E.xph /\ E*xE.yph) <-> (E.yE.xph /\ E*xE.yph))
5	4	biimpr 134	. . . 4 |- ((E.yE.xph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
6		excom 728	. . . 4 |- (E.xE.yph <-> E.yE.xph)
7	5, 6	sylanb 344	. . 3 |- ((E.xE.yph /\ E*xE.yph) -> E.y(E.xph /\ E*xE.yph))
8		2moex 1060	. . . . . . 7 |- (E*xE.yph -> A.yE*xph)
9	8	19.21bi 742	. . . . . 6 |- (E*xE.yph -> E*xph)
10	9	anim2i 270	. . . . 5 |- ((E.xph /\ E*xE.yph) -> (E.xph /\ E*xph))
11		eu5 1035	. . . . 5 |- (E!xph <-> (E.xph /\ E*xph))
12	10, 11	sylibr 175	. . . 4 |- ((E.xph /\ E*xE.yph) -> E!xph)
13	12	19.22i 723	. . 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 174	1 |- (E!xE.yph -> E.yE!xph)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678  E!weu 1007  E*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org