Theorem 2exeu 1489

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 1057	. . . . . . . 8 |- (E.xph -> A.xE.xph)
2	1	hbmo 1449	. . . . . . 7 |- (E*yE.xph -> A.xE*yE.xph)
3	2	19.41 1136	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) <-> (E.xE.yph /\ E*yE.xph))
4		19.8a 1070	. . . . . . . . 9 |- (ph -> E.xph)
5	4	immoi 1460	. . . . . . . 8 |- (E*yE.xph -> E*yph)
6	5	anim2i 342	. . . . . . 7 |- ((E.yph /\ E*yE.xph) -> (E.yph /\ E*yph))
7	6	19.22i 1081	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
8	3, 7	sylbir 208	. . . . 5 |- ((E.xE.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
9		excom 1087	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
10	8, 9	sylanb 460	. . . 4 |- ((E.yE.xph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
11		pm3.26 326	. . . . . 6 |- ((E.yph /\ E*yph) -> E.yph)
12	11	immoi 1460	. . . . 5 |- (E*xE.yph -> E*x(E.yph /\ E*yph))
13	12	adantl 397	. . . 4 |- ((E.xE.yph /\ E*xE.yph) -> E*x(E.yph /\ E*yph))
14	10, 13	anim12i 340	. . 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)))
15	14	ancoms 447	. 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)))
16		eu5 1451	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
17		eu5 1451	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
18	16, 17	anbi12i 493	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
19		eu5 1451	. . 3 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
20		eu5 1451	. . . . 5 |- (E!yph <-> (E.yph /\ E*yph))
21	20	exbii 1092	. . . 4 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
22	20	mobii 1447	. . . 4 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
23	21, 22	anbi12i 493	. . 3 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
24	19, 23	bitri 180	. 2 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
25	15, 18, 24	3imtr4i 226	1 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)

Syntax hints:   -> wi 3   /\ wa 230  E.wex 1021  E!weu 1422  E*wmo 1423

This theorem is referenced by:  2eu1 1492  2eu2 1493  2eu3 1494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425

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