Theorem 2eu1 1690

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 1642	. . . . . . . 8 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
2		eu5 1642	. . . . . . . . . 10 |- (E!yph <-> (E.yph /\ E*yph))
3	2	exbii 1236	. . . . . . . . 9 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
4	2	mobii 1638	. . . . . . . . 9 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
5	3, 4	anbi12i 537	. . . . . . . 8 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
6	1, 5	bitri 189	. . . . . . 7 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
7	6	pm3.27bi 351	. . . . . 6 |- (E!xE!yph -> E*x(E.yph /\ E*yph))
8		ax-4 1157	. . . . . . . . . . . 12 |- (A.xE*yph -> E*yph)
9	8	anim2i 360	. . . . . . . . . . 11 |- ((E.yph /\ A.xE*yph) -> (E.yph /\ E*yph))
10	9	ancoms 482	. . . . . . . . . 10 |- ((A.xE*yph /\ E.yph) -> (E.yph /\ E*yph))
11	10	immoi 1651	. . . . . . . . 9 |- (E*x(E.yph /\ E*yph) -> E*x(A.xE*yph /\ E.yph))
12		hba1 1188	. . . . . . . . . 10 |- (A.xE*yph -> A.xA.xE*yph)
13	12	moanim 1663	. . . . . . . . 9 |- (E*x(A.xE*yph /\ E.yph) <-> (A.xE*yph -> E*xE.yph))
14	11, 13	sylib 214	. . . . . . . 8 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> E*xE.yph))
15	14	ancrd 321	. . . . . . 7 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ A.xE*yph)))
16		2moswap 1685	. . . . . . . . 9 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
17	16	com12 14	. . . . . . . 8 |- (E*xE.yph -> (A.xE*yph -> E*yE.xph))
18	17	imdistani 489	. . . . . . 7 |- ((E*xE.yph /\ A.xE*yph) -> (E*xE.yph /\ E*yE.xph))
19	15, 18	syl6 25	. . . . . 6 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
20	7, 19	syl 12	. . . . 5 |- (E!xE!yph -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
21		2eu2ex 1684	. . . . . 6 |- (E!xE!yph -> E.xE.yph)
22		excom 1231	. . . . . . 7 |- (E.xE.yph <-> E.yE.xph)
23	21, 22	sylib 214	. . . . . 6 |- (E!xE!yph -> E.yE.xph)
24	21, 23	jca 308	. . . . 5 |- (E!xE!yph -> (E.xE.yph /\ E.yE.xph))
25	20, 24	jctild 659	. . . 4 |- (E!xE!yph -> (A.xE*yph -> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph))))
26		eu5 1642	. . . . . 6 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
27		eu5 1642	. . . . . 6 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
28	26, 27	anbi12i 537	. . . . 5 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
29		an4 561	. . . . 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	bitri 189	. . . 4 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph)))
31	25, 30	syl6ibr 229	. . 3 |- (E!xE!yph -> (A.xE*yph -> (E!xE.yph /\ E!yE.xph)))
32	31	com12 14	. 2 |- (A.xE*yph -> (E!xE!yph -> (E!xE.yph /\ E!yE.xph)))
33		2exeu 1687	. 2 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
34	32, 33	impbid1 572	1 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))

Syntax hints:   -> wi 3   <-> wb 162   /\ wa 239  A.wal 1134  E.wex 1164  E!weu 1609  E*wmo 1610

This theorem is referenced by:  2eu2 1691  2eu3 1692  2eu5 1694

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614

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