Theorem 2eu1 2284

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)

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 2242	. . . . . . . 8 |- (E!xE!yph <-> (E.xE!yph /\ E*xE!yph))
2		eu5 2242	. . . . . . . . . 10 |- (E!yph <-> (E.yph /\ E*yph))
3	2	exbii 1582	. . . . . . . . 9 |- (E.xE!yph <-> E.x(E.yph /\ E*yph))
4	2	mobii 2240	. . . . . . . . 9 |- (E*xE!yph <-> E*x(E.yph /\ E*yph))
5	3, 4	anbi12i 678	. . . . . . . 8 |- ((E.xE!yph /\ E*xE!yph) <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
6	1, 5	bitri 240	. . . . . . 7 |- (E!xE!yph <-> (E.x(E.yph /\ E*yph) /\ E*x(E.yph /\ E*yph)))
7	6	simprbi 450	. . . . . 6 |- (E!xE!yph -> E*x(E.yph /\ E*yph))
8		sp 1747	. . . . . . . . . . . 12 |- (A.xE*yph -> E*yph)
9	8	anim2i 552	. . . . . . . . . . 11 |- ((E.yph /\ A.xE*yph) -> (E.yph /\ E*yph))
10	9	ancoms 439	. . . . . . . . . 10 |- ((A.xE*yph /\ E.yph) -> (E.yph /\ E*yph))
11	10	moimi 2251	. . . . . . . . 9 |- (E*x(E.yph /\ E*yph) -> E*x(A.xE*yph /\ E.yph))
12		nfa1 1788	. . . . . . . . . 10 |- F/xA.xE*yph
13	12	moanim 2260	. . . . . . . . 9 |- (E*x(A.xE*yph /\ E.yph) <-> (A.xE*yph -> E*xE.yph))
14	11, 13	sylib 188	. . . . . . . 8 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> E*xE.yph))
15	14	ancrd 537	. . . . . . 7 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ A.xE*yph)))
16		2moswap 2279	. . . . . . . . 9 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
17	16	com12 27	. . . . . . . 8 |- (E*xE.yph -> (A.xE*yph -> E*yE.xph))
18	17	imdistani 671	. . . . . . 7 |- ((E*xE.yph /\ A.xE*yph) -> (E*xE.yph /\ E*yE.xph))
19	15, 18	syl6 29	. . . . . 6 |- (E*x(E.yph /\ E*yph) -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
20	7, 19	syl 15	. . . . 5 |- (E!xE!yph -> (A.xE*yph -> (E*xE.yph /\ E*yE.xph)))
21		2eu2ex 2278	. . . . . 6 |- (E!xE!yph -> E.xE.yph)
22		excom 1741	. . . . . . 7 |- (E.xE.yph <-> E.yE.xph)
23	21, 22	sylib 188	. . . . . 6 |- (E!xE!yph -> E.yE.xph)
24	21, 23	jca 518	. . . . 5 |- (E!xE!yph -> (E.xE.yph /\ E.yE.xph))
25	20, 24	jctild 527	. . . 4 |- (E!xE!yph -> (A.xE*yph -> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph))))
26		eu5 2242	. . . . . 6 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
27		eu5 2242	. . . . . 6 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
28	26, 27	anbi12i 678	. . . . 5 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E*xE.yph) /\ (E.yE.xph /\ E*yE.xph)))
29		an4 797	. . . . 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 240	. . . 4 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E.yE.xph) /\ (E*xE.yph /\ E*yE.xph)))
31	25, 30	syl6ibr 218	. . 3 |- (E!xE!yph -> (A.xE*yph -> (E!xE.yph /\ E!yE.xph)))
32	31	com12 27	. 2 |- (A.xE*yph -> (E!xE!yph -> (E!xE.yph /\ E!yE.xph)))
33		2exeu 2281	. 2 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
34	32, 33	impbid1 194	1 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  2eu2  2285  2eu3  2286  2eu5  2288