Theorem 2eu1 1426

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.

Assertion

Ref	Expression
2eu1	|- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))

Proof of Theorem 2eu1

Step	Hyp	Ref	Expression
1		eu5 1386	. . . . . . . 8 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
2		eu5 1386	. . . . . . . . . 10 |- (E!yph <-> (E.yph /\ E*yph))
3	2	exbii 1027	. . . . . . . . 9 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
4	2	mobii 1382	. . . . . . . . 9 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
5	3, 4	anbi12i 481	. . . . . . . 8 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
6	1, 5	bitr 173	. . . . . . 7 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
7	6	pm3.27bi 326	. . . . . 6 |- (E!xE!yph -> E*x(E.yph /\ E*yph))
8		ax-4 951	. . . . . . . . . . . 12 |- (A.xE*yph -> E*yph)
9	8	anim2i 335	. . . . . . . . . . 11 |- ((E.yph /\ A.xE*yph) -> (E.yph /\ E*yph))
10	9	ancoms 436	. . . . . . . . . 10 |- ((A.xE*yph /\ E.yph) -> (E.yph /\ E*yph))
11	10	immoi 1395	. . . . . . . . 9 |- (E*x(E.yph /\ E*yph) -> E*x(A.xE*yph /\ E.yph))
12		hba1 979	. . . . . . . . . 10 |- (A.xE*yph -> A.xA.xE*yph)
13	12	moanim 1404	. . . . . . . . 9 |- (E*x(A.xE*yph /\ E.yph) <-> (A.xE*yph -> E*xE.yph))
14	11, 13	sylib 198	. . . . . . . 8 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> E*xE.yph))
15	14	ancrd 299	. . . . . . 7 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ A.xE*yph)))
16		2moswap 1421	. . . . . . . . 9 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
17	16	com12 11	. . . . . . . 8 |- (E*xE.yph -> (A.xE*yph -> E*yE.xph))
18	17	imdistani 443	. . . . . . 7 |- ((E*xE.yph /\ A.xE*yph) -> (E*xE.yph /\ E*yE.xph))
19	15, 18	syl6 22	. . . . . 6 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
20	7, 19	syl 10	. . . . 5 |- (E!xE!yph -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
21		2eu2ex 1420	. . . . . 6 |- (E!xE!yph -> E.xE.yph)
22		excom 1022	. . . . . . 7 |- (E.xE.yph <-> E.yE.xph)
23	21, 22	sylib 198	. . . . . 6 |- (E!xE!yph -> E.yE.xph)
24	21, 23	jca 288	. . . . 5 |- (E!xE!yph -> (E.xE.yph /\ E.yE.xph))
25	20, 24	jctild 599	. . . 4 |- (E!xE!yph -> (A.xE*yph -> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph))))
26		eu5 1386	. . . . . 6 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
27		eu5 1386	. . . . . 6 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
28	26, 27	anbi12i 481	. . . . 5 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
29		an4 505	. . . . 5 |- (((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)) <-> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph)))
30	28, 29	bitr 173	. . . 4 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph)))
31	25, 30	syl6ibr 213	. . 3 |- (E!xE!yph -> (A.xE*yph -> (E!xE.yph /\ E!yE.xph)))
32	31	com12 11	. 2 |- (A.xE*yph -> (E!xE!yph -> (E!xE.yph /\ E!yE.xph)))
33		2exeu 1423	. 2 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
34	32, 33	impbid1 515	1 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956  E!weu 1357  E*wmo 1358

This theorem is referenced by:  2eu2 1427  2eu3 1428  2eu5 1430

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360

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