Theorem moexex 1415

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 1383	. . . . 5 |- (E*xph -> A.xE*xph)
2		hba1 979	. . . . . 6 |- (A.xE*yps -> A.xA.xE*yps)
3		hbe1 990	. . . . . . 7 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
4	3	hbmo 1384	. . . . . 6 |- (E*yE.x(ph /\ ps) -> A.xE*yE.x(ph /\ ps))
5	2, 4	hbim 983	. . . . 5 |- ((A.xE*yps -> E*yE.x(ph /\ ps)) -> A.x(A.xE*yps -> E*yE.x(ph /\ ps)))
6	1, 5	hbim 983	. . . 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 1384	. . . . . 6 |- (E*xph -> A.yE*xph)
9		mopick 1410	. . . . . . . 8 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
10	9	ex 373	. . . . . . 7 |- (E*xph -> (E.x(ph /\ ps) -> (ph -> ps)))
11	10	com3r 35	. . . . . 6 |- (ph -> (E*xph -> (E.x(ph /\ ps) -> ps)))
12	7, 8, 11	19.21ad 1035	. . . . 5 |- (ph -> (E*xph -> A.y(E.x(ph /\ ps) -> ps)))
13		immo 1394	. . . . . 6 |- (A.y(E.x(ph /\ ps) -> ps) -> (E*yps -> E*yE.x(ph /\ ps)))
14	13	a4sd 961	. . . . 5 |- (A.y(E.x(ph /\ ps) -> ps) -> (A.xE*yps -> E*yE.x(ph /\ ps)))
15	12, 14	syl6 22	. . . 4 |- (ph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
16	6, 15	19.23ai 1040	. . 3 |- (E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
17	7	hbex 982	. . . . . . . 8 |- (E.xph -> A.yE.xph)
18		pm3.26 319	. . . . . . . . 9 |- ((ph /\ ps) -> ph)
19	18	19.22i 1016	. . . . . . . 8 |- (E.x(ph /\ ps) -> E.xph)
20	17, 19	19.23ai 1040	. . . . . . 7 |- (E.yE.x(ph /\ ps) -> E.xph)
21	20	con3i 98	. . . . . 6 |- (-. E.xph -> -. E.yE.x(ph /\ ps))
22		exmo 1393	. . . . . . 7 |- (E.yE.x(ph /\ ps) \/ E*yE.x(ph /\ ps))
23	22	ori 230	. . . . . 6 |- (-. E.yE.x(ph /\ ps) -> E*yE.x(ph /\ ps))
24	21, 23	syl 10	. . . . 5 |- (-. E.xph -> E*yE.x(ph /\ ps))
25	24	a1d 12	. . . 4 |- (-. E.xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
26	25	a1d 12	. . 3 |- (-. E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
27	16, 26	pm2.61i 126	. 2 |- (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
28	27	imp 350	1 |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 950  E.wex 956  E*wmo 1358

This theorem is referenced by:  moexexv 1416  2moswap 1421

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360

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