Theorem mosubopt 3551

Description: "At most one" remains true inside ordered pair quantification.

Assertion

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

Distinct variable group:   x,y,z,A

Proof of Theorem mosubopt

Step	Hyp	Ref	Expression
1		hba1 1350	. . 3 |- (A.yA.zE*xph -> A.yA.yA.zE*xph)
2		hbe1 1363	. . . 4 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.yE.yE.z(A = <.y, z>. /\ ph))
3	2	hbmo 1803	. . 3 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.yE*xE.yE.z(A = <.y, z>. /\ ph))
4		hba1 1350	. . . . 5 |- (A.zE*xph -> A.zA.zE*xph)
5		hbe1 1363	. . . . . . 7 |- (E.z(A = <.y, z>. /\ ph) -> A.zE.z(A = <.y, z>. /\ ph))
6	5	hbex 1353	. . . . . 6 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.zE.yE.z(A = <.y, z>. /\ ph))
7	6	hbmo 1803	. . . . 5 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.zE*xE.yE.z(A = <.y, z>. /\ ph))
8		ax-17 1317	. . . . . . . 8 |- (A = <.y, z>. -> A.x A = <.y, z>.)
9		copsexg 3537	. . . . . . . 8 |- (A = <.y, z>. -> (ph <-> E.yE.z(A = <.y, z>. /\ ph)))
10	8, 9	mobid 1800	. . . . . . 7 |- (A = <.y, z>. -> (E*xph <-> E*xE.yE.z(A = <.y, z>. /\ ph)))
11	10	biimpcd 172	. . . . . 6 |- (E*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
12	11	a4s 1330	. . . . 5 |- (A.zE*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
13	4, 7, 12	19.23ad 1415	. . . 4 |- (A.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
14	13	a4s 1330	. . 3 |- (A.yA.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
15	1, 3, 14	19.23ad 1415	. 2 |- (A.yA.zE*xph -> (E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
16		simpl 346	. . . . . 6 |- ((A = <.y, z>. /\ ph) -> A = <.y, z>.)
17	16	2eximi 1388	. . . . 5 |- (E.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
18	17	19.23aiv 1674	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
19	18	con3i 114	. . 3 |- (-. E.yE.z A = <.y, z>. -> -. E.xE.yE.z(A = <.y, z>. /\ ph))
20		exmo 1812	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) \/ E*xE.yE.z(A = <.y, z>. /\ ph))
21	20	ori 247	. . 3 |- (-. E.xE.yE.z(A = <.y, z>. /\ ph) -> E*xE.yE.z(A = <.y, z>. /\ ph))
22	19, 21	syl 12	. 2 |- (-. E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
23	15, 22	pm2.61d1 142	1 |- (A.yA.zE*xph -> E*xE.yE.z(A = <.y, z>. /\ ph))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  E*wmo 1772  <.cop 3046

This theorem is referenced by:  mosubop 3552  funoprabg 4939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

Copyright terms: Public domain