Theorem moexex 1058

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 1032	. . . . 5 |- (E*xph -> A.xE*xph)
2		hba1 698	. . . . . 6 |- (A.xE*yps -> A.xA.xE*yps)
3		hbe1 709	. . . . . . 7 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
4	3	hbmo 1033	. . . . . 6 |- (E*yE.x(ph /\ ps) -> A.xE*yE.x(ph /\ ps))
5	2, 4	hbim 702	. . . . 5 |- ((A.xE*yps -> E*yE.x(ph /\ ps)) -> A.x(A.xE*yps -> E*yE.x(ph /\ ps)))
6	1, 5	hbim 702	. . . 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 1033	. . . . . 6 |- (E*xph -> A.yE*xph)
9		mopick 1054	. . . . . . . 8 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
10	9	exp 291	. . . . . . 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 741	. . . . 5 |- (ph -> (E*xph -> A.y(E.x(ph /\ ps) -> ps)))
13		immo 1043	. . . . . 6 |- (A.y(E.x(ph /\ ps) -> ps) -> (E*yps -> E*yE.x(ph /\ ps)))
14	13	a4sd 683	. . . . 5 |- (A.y(E.x(ph /\ ps) -> ps) -> (A.xE*yps -> E*yE.x(ph /\ ps)))
15	12, 14	syl6 23	. . . 4 |- (ph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
16	6, 15	19.23ai 746	. . 3 |- (E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
17	7	hbex 701	. . . . . . . 8 |- (E.xph -> A.yE.xph)
18		pm3.26 256	. . . . . . . . 9 |- ((ph /\ ps) -> ph)
19	18	19.22i 723	. . . . . . . 8 |- (E.x(ph /\ ps) -> E.xph)
20	17, 19	19.23ai 746	. . . . . . 7 |- (E.yE.x(ph /\ ps) -> E.xph)
21	20	con3i 90	. . . . . 6 |- (-. E.xph -> -. E.yE.x(ph /\ ps))
22		exmo 1042	. . . . . . 7 |- (E.yE.x(ph /\ ps) \/ E*yE.x(ph /\ ps))
23	22	ori 200	. . . . . 6 |- (-. E.yE.x(ph /\ ps) -> E*yE.x(ph /\ ps))
24	21, 23	syl 12	. . . . 5 |- (-. E.xph -> E*yE.x(ph /\ ps))
25	24	a1d 14	. . . 4 |- (-. E.xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
26	25	a1d 14	. . 3 |- (-. E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
27	16, 26	pm2.61i 110	. 2 |- (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
28	27	imp 277	1 |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678  E*wmo 1008

This theorem is referenced by:  moexexv 1059  2moswap 1064

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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