Theorem 2exeu 1066

Description: Double existential uniqueness implies double uniqueness quantification.

Assertion

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

Proof of Theorem 2exeu

Step	Hyp	Ref	Expression
1		hbe1 709	. . . . . . . 8 |- (E.xph -> A.xE.xph)
2	1	hbmo 1033	. . . . . . 7 |- (E*yE.xph -> A.xE*yE.xph)
3	2	19.41 774	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) <-> (E.xE.yph /\ E*yE.xph))
4		immo 1043	. . . . . . . . 9 |- (A.y(ph -> E.xph) -> (E*yE.xph -> E*yph))
5		19.8a 712	. . . . . . . . 9 |- (ph -> E.xph)
6	4, 5	mpg 684	. . . . . . . 8 |- (E*yE.xph -> E*yph)
7	6	anim2i 270	. . . . . . 7 |- ((E.yph /\ E*yE.xph) -> (E.yph /\ E*yph))
8	7	19.22i 723	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
9	3, 8	sylbir 176	. . . . 5 |- ((E.xE.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
10		excom 728	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
11	9, 10	sylanb 344	. . . 4 |- ((E.yE.xph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
12		immo 1043	. . . . . 6 |- (A.x((E.yph /\ E*yph) -> E.yph) -> (E*xE.yph -> E*x(E.yph /\ E*yph)))
13		pm3.26 256	. . . . . 6 |- ((E.yph /\ E*yph) -> E.yph)
14	12, 13	mpg 684	. . . . 5 |- (E*xE.yph -> E*x(E.yph /\ E*yph))
15	14	adantl 305	. . . 4 |- ((E.xE.yph /\ E*xE.yph) -> E*x(E.yph /\ E*yph))
16	11, 15	anim12i 268	. . 3 |- (((E.yE.xph /\ E*yE.xph) /\ (E.xE.yph /\ E*xE.yph)) -> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
17	16	ancoms 334	. 2 |- (((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)) -> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
18		eu5 1035	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
19		eu5 1035	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
20	18, 19	anbi12i 369	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
21		eu5 1035	. . 3 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
22		eu5 1035	. . . . 5 |- (E!yph <-> (E.yph /\ E*yph))
23	22	biex 733	. . . 4 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
24	22	bimo 1031	. . . 4 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
25	23, 24	anbi12i 369	. . 3 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
26	21, 25	bitr 151	. 2 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
27	17, 20, 26	3imtr4 192	1 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)

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

This theorem is referenced by:  2eu1 1067  2eu2 1068  2eu3 1069

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

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