Theorem mosubopt 2799

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 1001	. . 3 |- (A.yA.zE*xph -> A.yA.yA.zE*xph)
2		hbe1 1014	. . . 4 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.yE.yE.z(A = <.y, z>. /\ ph))
3	2	hbmo 1405	. . 3 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.yE*xE.yE.z(A = <.y, z>. /\ ph))
4		hba1 1001	. . . . 5 |- (A.zE*xph -> A.zA.zE*xph)
5		hbe1 1014	. . . . . . 7 |- (E.z(A = <.y, z>. /\ ph) -> A.zE.z(A = <.y, z>. /\ ph))
6	5	hbex 1004	. . . . . 6 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.zE.yE.z(A = <.y, z>. /\ ph))
7	6	hbmo 1405	. . . . 5 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.zE*xE.yE.z(A = <.y, z>. /\ ph))
8		ax-17 969	. . . . . . . 8 |- (A = <.y, z>. -> A.x A = <.y, z>.)
9		copsexg 2787	. . . . . . . 8 |- (A = <.y, z>. -> (ph <-> E.yE.z(A = <.y, z>. /\ ph)))
10	8, 9	mobid 1402	. . . . . . 7 |- (A = <.y, z>. -> (E*xph <-> E*xE.yE.z(A = <.y, z>. /\ ph)))
11	10	biimpcd 155	. . . . . 6 |- (E*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
12	11	a4s 982	. . . . 5 |- (A.zE*xph -> (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
13	4, 7, 12	19.23ad 1064	. . . 4 |- (A.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
14	13	a4s 982	. . 3 |- (A.yA.zE*xph -> (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
15	1, 3, 14	19.23ad 1064	. 2 |- (A.yA.zE*xph -> (E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph)))
16		pm3.26 319	. . . . . 6 |- ((A = <.y, z>. /\ ph) -> A = <.y, z>.)
17	16	19.22i2 1039	. . . . 5 |- (E.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
18	17	19.23aiv 1293	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
19	18	con3i 98	. . 3 |- (-. E.yE.z A = <.y, z>. -> -. E.xE.yE.z(A = <.y, z>. /\ ph))
20		exmo 1414	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) \/ E*xE.yE.z(A = <.y, z>. /\ ph))
21	20	ori 230	. . 3 |- (-. E.xE.yE.z(A = <.y, z>. /\ ph) -> E*xE.yE.z(A = <.y, z>. /\ ph))
22	19, 21	syl 10	. 2 |- (-. E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
23	15, 22	pm2.61d1 128	1 |- (A.yA.zE*xph -> E*xE.yE.z(A = <.y, z>. /\ ph))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 952   = wceq 954  E.wex 978  E*wmo 1379  <.cop 2407

This theorem is referenced by:  mosubop 2800  funoprabg 4001

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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