Theorem moexex 1841

Description: "At most one" double quantification.

Hypothesis

Ref	Expression
moexex.1	|- (ph -> A.yph)

Assertion

Ref	Expression
moexex	|- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Proof of Theorem moexex

Step	Hyp	Ref	Expression
1		hbmo1 1802	. . . . 5 |- (E*xph -> A.xE*xph)
2		hba1 1350	. . . . . 6 |- (A.xE*yps -> A.xA.xE*yps)
3		hbe1 1363	. . . . . . 7 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
4	3	hbmo 1803	. . . . . 6 |- (E*yE.x(ph /\ ps) -> A.xE*yE.x(ph /\ ps))
5	2, 4	hbim 1354	. . . . 5 |- ((A.xE*yps -> E*yE.x(ph /\ ps)) -> A.x(A.xE*yps -> E*yE.x(ph /\ ps)))
6	1, 5	hbim 1354	. . . 4 |- ((E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))) -> A.x(E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
7		moexex.1	. . . . . 6 |- (ph -> A.yph)
8	7	hbmo 1803	. . . . . 6 |- (E*xph -> A.yE*xph)
9		mopick 1833	. . . . . . . 8 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
10	9	ex 402	. . . . . . 7 |- (E*xph -> (E.x(ph /\ ps) -> (ph -> ps)))
11	10	com3r 39	. . . . . 6 |- (ph -> (E*xph -> (E.x(ph /\ ps) -> ps)))
12	7, 8, 11	19.21ad 1406	. . . . 5 |- (ph -> (E*xph -> A.y(E.x(ph /\ ps) -> ps)))
13		immo 1813	. . . . . 6 |- (A.y(E.x(ph /\ ps) -> ps) -> (E*yps -> E*yE.x(ph /\ ps)))
14	13	a4sd 1331	. . . . 5 |- (A.y(E.x(ph /\ ps) -> ps) -> (A.xE*yps -> E*yE.x(ph /\ ps)))
15	12, 14	syl6 25	. . . 4 |- (ph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
16	6, 15	19.23ai 1412	. . 3 |- (E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
17	7	hbex 1353	. . . . . . . 8 |- (E.xph -> A.yE.xph)
18		exsimpl 1461	. . . . . . . 8 |- (E.x(ph /\ ps) -> E.xph)
19	17, 18	19.23ai 1412	. . . . . . 7 |- (E.yE.x(ph /\ ps) -> E.xph)
20	19	con3i 114	. . . . . 6 |- (-. E.xph -> -. E.yE.x(ph /\ ps))
21		exmo 1812	. . . . . . 7 |- (E.yE.x(ph /\ ps) \/ E*yE.x(ph /\ ps))
22	21	ori 247	. . . . . 6 |- (-. E.yE.x(ph /\ ps) -> E*yE.x(ph /\ ps))
23	20, 22	syl 12	. . . . 5 |- (-. E.xph -> E*yE.x(ph /\ ps))
24	23	a1d 15	. . . 4 |- (-. E.xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
25	24	a1d 15	. . 3 |- (-. E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
26	16, 25	pm2.61i 140	. 2 |- (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
27	26	imp 377	1 |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296  E.wex 1326  E*wmo 1772

This theorem is referenced by:  moexexv 1842  2moswap 1848

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-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

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

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