Theorem 2exeu 1446

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 1016	. . . . . . . 8 |- (E.xph -> A.xE.xph)
2	1	hbmo 1407	. . . . . . 7 |- (E*yE.xph -> A.xE*yE.xph)
3	2	19.41 1095	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) <-> (E.xE.yph /\ E*yE.xph))
4		19.8a 1029	. . . . . . . . 9 |- (ph -> E.xph)
5	4	immoi 1418	. . . . . . . 8 |- (E*yE.xph -> E*yph)
6	5	anim2i 335	. . . . . . 7 |- ((E.yph /\ E*yE.xph) -> (E.yph /\ E*yph))
7	6	19.22i 1040	. . . . . 6 |- (E.x(E.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
8	3, 7	sylbir 201	. . . . 5 |- ((E.xE.yph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
9		excom 1046	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
10	8, 9	sylanb 449	. . . 4 |- ((E.yE.xph /\ E*yE.xph) -> E.x(E.yph /\ E*yph))
11		pm3.26 319	. . . . . 6 |- ((E.yph /\ E*yph) -> E.yph)
12	11	immoi 1418	. . . . 5 |- (E*xE.yph -> E*x(E.yph /\ E*yph))
13	12	adantl 388	. . . 4 |- ((E.xE.yph /\ E*xE.yph) -> E*x(E.yph /\ E*yph))
14	10, 13	anim12i 333	. . 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 436	. 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 1409	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
17		eu5 1409	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
18	16, 17	anbi12i 482	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
19		eu5 1409	. . 3 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
20		eu5 1409	. . . . 5 |- (E!yph <-> (E.yph /\ E*yph))
21	20	exbii 1051	. . . 4 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
22	20	mobii 1405	. . . 4 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
23	21, 22	anbi12i 482	. . 3 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
24	19, 23	bitr 173	. 2 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
25	15, 18, 24	3imtr4 219	1 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)

Syntax hints:   -> wi 3   /\ wa 223  E.wex 980  E!weu 1380  E*wmo 1381

This theorem is referenced by:  2eu1 1449  2eu2 1450  2eu3 1451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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