Theorem mosubopt 4613

Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Assertion

Ref	Expression
mosubopt	|- (A.yA.zE*xph -> E*xE.yE.z(A = <.y, z>. /\ ph))

Distinct variable group:   x,y,z,A

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem mosubopt

Step	Hyp	Ref	Expression
1		nfa1 1788	. . 3 |- F/yA.yA.zE*xph
2		nfe1 1732	. . . 4 |- F/yE.yE.z(A = <.y, z>. /\ ph)
3	2	nfmo 2221	. . 3 |- F/yE*xE.yE.z(A = <.y, z>. /\ ph)
4		nfa1 1788	. . . . 5 |- F/zA.zE*xph
5		nfe1 1732	. . . . . . 7 |- F/zE.z(A = <.y, z>. /\ ph)
6	5	nfex 1843	. . . . . 6 |- F/zE.yE.z(A = <.y, z>. /\ ph)
7	6	nfmo 2221	. . . . 5 |- F/zE*xE.yE.z(A = <.y, z>. /\ ph)
8		copsexg 4608	. . . . . . . 8 |- (A = <.y, z>. -> (ph <-> E.yE.z(A = <.y, z>. /\ ph)))
9	8	mobidv 2239	. . . . . . 7 |- (A = <.y, z>. -> (E*xph <-> E*xE.yE.z(A = <.y, z>. /\ ph)))
10	9	biimpcd 215	. . . . . 6 |- (E*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
11	10	sps 1754	. . . . 5 |- (A.zE*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
12	4, 7, 11	exlimd 1806	. . . 4 |- (A.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
13	12	sps 1754	. . 3 |- (A.yA.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
14	1, 3, 13	exlimd 1806	. 2 |- (A.yA.zE*xph -> (E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
15		simpl 443	. . . . . 6 |- ((A = <.y, z>. /\ ph) -> A = <.y, z>.)
16	15	2eximi 1577	. . . . 5 |- (E.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
17	16	exlimiv 1634	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
18	17	con3i 127	. . 3 |- (-. E.yE.z A = <.y, z>. -> -. E.xE.yE.z(A = <.y, z>. /\ ph))
19		exmo 2249	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) \/ E*xE.yE.z(A = <.y, z>. /\ ph))
20	19	ori 364	. . 3 |- (-. E.xE.yE.z(A = <.y, z>. /\ ph) -> E*xE.yE.z(A = <.y, z>. /\ ph))
21	18, 20	syl 15	. 2 |- (-. E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
22	14, 21	pm2.61d1 151	1 |- (A.yA.zE*xph -> E*xE.yE.z(A = <.y, z>. /\ ph))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  E*wmo 2205  <.cop 4562

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  mosubop  4614  funoprabg  5584